Euclidean vector - Knowledge

9722: 9274: 2049: 11711:. The vector itself does not change when the basis is transformed; instead, the components of the vector make a change that cancels the change in the basis. In other words, if the reference axes (and the basis derived from it) were rotated in one direction, the component representation of the vector would rotate in the opposite way to generate the same final vector. Similarly, if the reference axes were stretched in one direction, the components of the vector would reduce in an exactly compensating way. Mathematically, if the basis undergoes a transformation described by an 8756: 13063: 5591: 5388: 10553: 8401: 9717:{\displaystyle {\begin{aligned}c_{11}&=\mathbf {n} _{1}\cdot \mathbf {e} _{1}\\c_{12}&=\mathbf {n} _{1}\cdot \mathbf {e} _{2}\\c_{13}&=\mathbf {n} _{1}\cdot \mathbf {e} _{3}\\c_{21}&=\mathbf {n} _{2}\cdot \mathbf {e} _{1}\\c_{22}&=\mathbf {n} _{2}\cdot \mathbf {e} _{2}\\c_{23}&=\mathbf {n} _{2}\cdot \mathbf {e} _{3}\\c_{31}&=\mathbf {n} _{3}\cdot \mathbf {e} _{1}\\c_{32}&=\mathbf {n} _{3}\cdot \mathbf {e} _{2}\\c_{33}&=\mathbf {n} _{3}\cdot \mathbf {e} _{3}\end{aligned}}} 5362: 1728: 3596: 5028: 6688: 10109: 5912: 13327: 42: 8392: 11566: 2072: 8751:{\displaystyle {\begin{aligned}u&=p\mathbf {e} _{1}\cdot \mathbf {n} _{1}+q\mathbf {e} _{2}\cdot \mathbf {n} _{1}+r\mathbf {e} _{3}\cdot \mathbf {n} _{1},\\v&=p\mathbf {e} _{1}\cdot \mathbf {n} _{2}+q\mathbf {e} _{2}\cdot \mathbf {n} _{2}+r\mathbf {e} _{3}\cdot \mathbf {n} _{2},\\w&=p\mathbf {e} _{1}\cdot \mathbf {n} _{3}+q\mathbf {e} _{2}\cdot \mathbf {n} _{3}+r\mathbf {e} _{3}\cdot \mathbf {n} _{3}.\end{aligned}}} 11525: 7529: 6159: 2084: 11632: 8109: 10548:{\displaystyle {\begin{aligned}(a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}+a_{4}{\mathbf {e} }_{4})&+(b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}+b_{4}{\mathbf {e} }_{4})=\\(a_{1}+b_{1}){\mathbf {e} }_{1}+(a_{2}+b_{2}){\mathbf {e} }_{2}&+(a_{3}+b_{3}){\mathbf {e} }_{3}+(a_{4}+b_{4}){\mathbf {e} }_{4}.\end{aligned}}} 1670:, depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as 7328: 5984: 9186: 10104: 10779: 6990: 8043: 7790: 7321: 8387:{\displaystyle {\begin{aligned}u&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{1},\\v&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{2},\\w&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{3}.\end{aligned}}} 11804:

is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an

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of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of

519:. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of a special kind of vector space called 1662:. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called 8075:

in both cases. It is common to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In such a case it is necessary to develop a method to convert between bases so the basic vector operations such as addition

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length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of

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of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector

3149: 8967: 11250: 3537: 7524:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=(\mathbf {c} \ \mathbf {a} \ \mathbf {b} )=(\mathbf {b} \ \mathbf {c} \ \mathbf {a} )=-(\mathbf {a} \ \mathbf {c} \ \mathbf {b} )=-(\mathbf {b} \ \mathbf {a} \ \mathbf {c} )=-(\mathbf {c} \ \mathbf {b} \ \mathbf {a} ).} 5298: 4983: 388:

The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the

6628: 6154:{\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}} 9880: 1713:. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction". 10571: 6760: 7916: 7663: 6352: 7153: 7111: 417:(Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s. 5552: 11014: 4123: 2639: 4669: 4560: 4340: 4235: 2795: 4011: 3793: 3003: 3730: 2996: 523:. This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as 6215:, or simply 0. Unlike any other vector, it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector that is a multiple of the zero vector). The sum of the zero vector with any vector 8765: 11080: 5902: 5813: 191:

means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of

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Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the

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This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector

11510: 5138: 4823: 3663: 3323: 6553: 2491: 11348: 9181:{\displaystyle {\begin{bmatrix}u\\v\\w\\\end{bmatrix}}={\begin{bmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{bmatrix}}{\begin{bmatrix}p\\q\\r\end{bmatrix}}.} 4780: 4438: 7123:, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors 661: 1562: 6286: 1418:. It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive 7039: 5936:

is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as

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The vector concept, as it is known today, is the result of a gradual development over a period of more than 200 years. About a dozen people contributed significantly to its development. In 1835,

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Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is

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acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example,

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of two vectors from the complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth.

1187: 809: 2218: 1678:) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1 1384: 933: 895: 10099:{\displaystyle (a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2})+(b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2})=(a_{1}+b_{1}){\mathbf {e} }_{1}+(a_{2}+b_{2}){\mathbf {e} }_{2},} 9877:

With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as

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By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.

10774:{\displaystyle (a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2})\wedge (b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2})=(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{1}\mathbf {e} _{2}.} 6985:{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2}){\mathbf {e} }_{1}+(a_{3}b_{1}-a_{1}b_{3}){\mathbf {e} }_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf {e} }_{3}.} 2184: 3353: 10825:

inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.

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is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.

8038:{\displaystyle {\begin{aligned}u&=\mathbf {a} \cdot \mathbf {n} _{1},\\v&=\mathbf {a} \cdot \mathbf {n} _{2},\\w&=\mathbf {a} \cdot \mathbf {n} _{3}.\end{aligned}}} 7785:{\displaystyle {\begin{aligned}p&=\mathbf {a} \cdot \mathbf {e} _{1},\\q&=\mathbf {a} \cdot \mathbf {e} _{2},\\r&=\mathbf {a} \cdot \mathbf {e} _{3}.\end{aligned}}} 2826: 1219: 1128: 1077: 1045: 1012: 1988: 7563:}. However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector. In the 5854: 1812: 1789: 1766: 978:. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. 836:). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to 7316:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}} 5727: 7576: 6209: 1855: 1411:

In Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin

7832: 275:), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the 6404: 1300:, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, 12432: 3739: 11809:, but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include 11774: 11442: 6532:

dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted

3676: 7024:) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by ( 6994:

For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a

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Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors

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of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines

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The decomposition or resolution of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.

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An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a

558:. When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a 12514: 4345: 3144:{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},} 11780:. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if 9866:

to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above.

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defined by two vectors (used as sides of the parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the

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has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the

11840:, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a 3609: 3269: 1429: 8962:{\displaystyle {\begin{aligned}u&=c_{11}p+c_{12}q+c_{13}r,\\v&=c_{21}p+c_{22}q+c_{23}r,\\w&=c_{31}p+c_{32}q+c_{33}r,\end{aligned}}} 11867:

gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the

11245:{\displaystyle {\mathbf {y} }-{\mathbf {x} }=(y_{1}-x_{1}){\mathbf {e} }_{1}+(y_{2}-x_{2}){\mathbf {e} }_{2}+(y_{3}-x_{3}){\mathbf {e} }_{3}.} 3669:

in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a

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However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of

3532:{\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.} 12879: 3805:

as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively

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angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include

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containing the information that relates the two bases. Such an expression can be formed by substitution of the above equations to form

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which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are

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is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ

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The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation

11396: 5293:{\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1})\mathbf {e} _{1}+(a_{2}-b_{2})\mathbf {e} _{2}+(a_{3}-b_{3})\mathbf {e} _{3}.} 4978:{\displaystyle \mathbf {a} +\mathbf {b} =(a_{1}+b_{1})\mathbf {e} _{1}+(a_{2}+b_{2})\mathbf {e} _{2}+(a_{3}+b_{3})\mathbf {e} _{3}.} 12921: 11972: 11849: 1683: 3795:). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. 6623:{\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} } 3844:

In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a

13254: 13312: 1340:, a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points 679:, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. 12472:

Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs

12845: 12821: 12716: 12672: 12646: 9809:" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from 1018:. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, 7142:), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the 3231: 619: 1575:

or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an

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A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the

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Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip

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on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane. The term

31: 6347:{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,} 1272:

would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless.

475:, adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus. 13264: 13200: 12835: 10839:

Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter

10784: 6529: 1604:, which (among other things) supplies an algebraic characterization of the area and orientation in space of the 11591: 7106:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).} 1870: 1727: 393:

Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including

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The advantage of this method is that a direction cosine matrix can usually be obtained independently by using

5027: 12872: 12737: 12518: 4452: 3670: 1157: 779: 2191: 1646:, where many quantities of interest can be considered vectors in a space with no notion of length or angle. 1357: 908: 870: 13371: 13042: 12914: 12321: 8071:

relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vector

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also has generalizations to higher dimensions, and to more formal approaches with much wider applications.

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This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying

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of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be

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is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.

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copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

5547:{\displaystyle r\mathbf {a} =(ra_{1})\mathbf {e} _{1}+(ra_{2})\mathbf {e} _{2}+(ra_{3})\mathbf {e} _{3}.} 3867: 3798:

The choice of a basis does not affect the properties of a vector or its behaviour under transformations.

3733: 2810: 2119: 2099: 1642:(for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from 1337: 1253: 663:

in space represent the same free vector if they have the same magnitude and direction: that is, they are

3331: 13381: 13052: 12946: 11009:{\displaystyle {\mathbf {x} }=x_{1}{\mathbf {e} }_{1}+x_{2}{\mathbf {e} }_{2}+x_{3}{\mathbf {e} }_{3}.} 4118:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.} 2634:{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=^{\operatorname {T} }.} 1244:

has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force

1192: 1101: 898: 10558: 4664:{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} 4555:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} 4335:{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} 4230:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} 2090:

In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an

1053: 1021: 988: 13356: 13292: 12941: 12163: 7007: 2790:{\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).} 1969: 1571:

In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, a

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by differentiating or integrating the components of the vector, and many of the familiar rules from

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The scalar triple product is linear in all three entries and anti-symmetric in the following sense:

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The dot product can also be defined as the sum of the products of the components of each vector as

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is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (

3849: 1139: 1083: 436: 9817:" (because it contains direction cosines). The properties of a rotation matrix are such that its 1795: 1772: 1749: 13366: 13330: 13259: 13037: 12907: 12168: 12153: 11580: 8101: 5683: 2042: 707: 664: 492: 465:, published in 1881, presents what is essentially the modern system of vector analysis. In 1901, 304: 244:

vectors as an example of the more generalized concept of vectors defined simply as elements of a

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All examples thus far have dealt with vectors expressed in terms of the same basis, namely, the

6185: 5418:) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called 1831: 13094: 13027: 13017: 12066: 11708: 11390: 10834: 6373: 3802: 3788:{\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}} 3666: 3567: 1620: 1593: 975: 320: 12545: 11873:

of space. A vector which gains a minus sign when the orientation of space changes is called a

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The cross product does not readily generalise to other dimensions, though the closely related

564:. The distinction between bound and free vectors is especially relevant in mechanics, where a 13109: 13104: 13099: 13032: 12977: 12176: 11837: 11810: 11544: 11437: 11074: 9786: 7115:

It has three primary uses. First, the absolute value of the box product is the volume of the

5382: 3725:{\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} } 1905: 1722: 1609: 1301: 1260:

is represented by the vector 15 N, and if positive points leftward, then the vector for

1147: 983: 500: 406: 272: 120: 115: 12862: 12727: 11980:, a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in 3861: 2048: 13119: 13084: 13071: 12962: 12467: 11981: 11750: 10817: 10810: 7147: 5074:

are bound vectors that have the same base point, this point will also be the base point of

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A vector can also be broken up with respect to "non-fixed" basis vectors that change their

3555: 2504: 2115: 1825: 1305: 837: 676: 512: 466: 450: 312: 111: 93: 11646: 4794:) if they have the same direction but not necessarily the same magnitude. Two vectors are 2991:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ } 1900:, which is a convention for indicating boldface type. If the vector represents a directed 8: 13351: 13297: 13177: 13152: 13002: 12893: 12020: 11869: 10856: 10813: 7120: 6265: 5818: 3834: 2060: 1706: 864: 454: 431: 308: 280: 205: 12504:

respectively. This resolution may be accomplished by constructing the parallelogram ..."

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the rows and columns are orthogonal unit vectors, therefore their dot products are zero.

1623:, a vector's squared length can be positive, negative, or zero. An important example is 13007: 12782: 12139: 11707:

is required to have components that "transform opposite to the basis" under changes of

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pointing into and behind the diagram. These can be thought of as viewing the tip of an

1628: 1324:, are represented as a system of vectors at each point of a physical space; that is, a 699: 695: 573: 496: 461:, separated off their vector part for independent treatment. The first half of Gibbs's 418: 228:

have close analogues for vectors, operations which obey the familiar algebraic laws of

13205: 13162: 13089: 12982: 12841: 12817: 12811: 12789: 12763: 12712: 12689: 12668: 12642: 12605: 12595: 12551: 12208: 11967: 11712: 11700: 5897:{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}.} 3581: 2095: 1710: 1229: 1143: 1135: 940: 703: 398: 359: 300: 276: 268: 107: 12667:. Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications. Wiley. 12641:. Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra. Wiley. 5808:{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}},} 5590: 5302:

Subtraction of two vectors can be geometrically illustrated as follows: to subtract

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if they have the same or opposite direction but not necessarily the same magnitude.

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to distinguish them from pseudovectors. Pseudovectors occur most frequently as the

9822: 7645:{\displaystyle \mathbf {a} =p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3}.} 3222: 3176: 2124: 1601: 1313: 508: 225: 12262: 7898:{\displaystyle \mathbf {a} =u\mathbf {n} _{1}+v\mathbf {n} _{2}+w\mathbf {n} _{3}} 2797:

These have the intuitive interpretation as vectors of unit length pointing up the

2045:, while the direction in which the arrow points indicates the vector's direction. 402: 13269: 13062: 13022: 13012: 12831: 12594:(3rd ed.). Reston, Va.: American Institute of Aeronautics and Astronautics. 12061: 12056: 10905: 10859:

representation of the trajectory of the particle. Vector-valued functions can be

10796: 9798: 6732: 6661: 5407: 5387: 4786: 1959: 1675: 1624: 1309: 1241: 569: 520: 504: 471: 394: 241: 102: 6496:{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.} 3228:

In introductory physics textbooks, the standard basis vectors are often denoted

1658:

of components, or list of numbers, that act as scalar coefficients for a set of

13274: 13195: 12930: 12456: 11986: 11962: 11919: 11818: 9818: 7116: 5718: 5706: 5610: 3551: 2648: 2065: 1643: 1584:

a vector by itself). In three dimensions, it is further possible to define the

1321: 1317: 902: 856: 367: 363: 237: 221: 89: 12333: 11505:{\displaystyle W={\mathbf {F} }\cdot ({\mathbf {x} }_{2}-{\mathbf {x} }_{1}).} 4013:

and assumes that all vectors have the origin as a common base point. A vector

1690:

are another type of quantity that behave in this way; a vector is one type of

311:

line segments of the same length and orientation. Essentially, he realized an

283:

and transform in a similar way under changes of the coordinate system include

13345: 13307: 13230: 13190: 13157: 13137: 12683: 12661: 12636: 12609: 12086:"Can be brought to the same straight line by means of parallel displacement". 12051: 12005: 12000: 11892: 11853: 10871:

continue to hold for the derivative and integral of vector-valued functions.

6649: 6511: 5055: 5000:

The addition may be represented graphically by placing the tail of the arrow

3838: 2496: 2056: 1597: 1585: 1292:

could be represented by the vector (0, 5) (in 2 dimensions with the positive

1249: 757: 672: 444: 233: 229: 12287: 5391:

Scalar multiplication of a vector by a factor of 3 stretches the vector out.

13240: 13129: 13079: 12972: 12777: 12025: 11952: 11943: 11875: 11830: 11806: 11368: 10566: 9859: 6995: 4807: 3658:{\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } 3359:). In this case, the scalar and vector components are denoted respectively 3318:{\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } 3188:

on the basis vectors or, equivalently, on the corresponding Cartesian axes

1994: 1702: 1659: 1639: 1635: 1325: 1269: 944: 849: 814: 711: 516: 284: 260: 245: 97: 6178:

is the vector with length zero. Written out in coordinates, the vector is

3599:

Illustration of tangential and normal components of a vector to a surface.

1993:

Vectors are usually shown in graphs or other diagrams as arrows (directed

491:, a vector is typically regarded as a geometric entity characterized by a 13220: 13185: 13142: 12987: 12656: 12632: 12046: 12010: 11977: 7143: 6664:

system. The right-handedness constraint is necessary because there exist

6246: 6169: 5927: 5846: 5400: 3595: 3356: 2135:

As an example in two dimensions (see figure), the vector from the origin

1698: 1580: 1095: 1015: 936: 488: 440: 342: 217: 209: 65: 57: 11856:, and the rules for transforming a contravariant vector follow from the 10821:

different dimension has no particular significance unless there is some

1654:

In physics, as well as mathematics, a vector is often identified with a

1090:. This is motivated by the fact that every Euclidean space of dimension 124:. A vector is frequently depicted graphically as an arrow connecting an 13249: 12992: 12755: 12474:, by E.B. Wilson, Chares Scribner's Sons, New York, p. 15: "Any vector 12030: 11857: 10860: 9863: 6728: 6687: 5911: 3326: 2500: 2486:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).} 1130:

More precisely, given such a Euclidean space, one may choose any point

324: 11343:{\displaystyle {\mathbf {x} }_{t}=t{\mathbf {v} }+{\mathbf {x} }_{0},} 1682:/m becomes 0.001 K/mm—a covariant change in value (for more, see 1189:

These choices define an isomorphism of the given Euclidean space onto

13047: 12708: 11991: 4775:{\displaystyle a_{1}=-b_{1},\quad a_{2}=-b_{2},\quad a_{3}=-b_{3}.\,} 3558:

commonly used in higher level mathematics, physics, and engineering.

2651:

vectors. For instance, in three dimensions, there are three of them:

2102:. The endpoint of a vector can be identified with an ordered list of 1962:

literature, it was especially common to represent vectors with small

1821: 1743: 371: 41: 11565: 9265:

of the angle between two unit vectors, which is also equal to their

13215: 12184: 11930: 11852:

rank one. Alternatively, a contravariant vector is defined to be a

11822: 11814: 11797: 11272: 10868: 10864: 10562: 8971:

and these equations can be expressed as the single matrix equation

5564:. Geometrically, this can be visualized (at least in the case when 3829: 1901: 1671: 1281: 841: 256: 213: 11260:. The length of this vector gives the straight-line distance from 4433:{\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,} 2071: 12899: 3576:, a vector is often described by a set of vector components that 2075:

A vector in the Cartesian plane, showing the position of a point

1963: 484: 252: 61: 10809:

In abstract vector spaces, the length of the arrow depends on a

358:. Like Bellavitis, Hamilton viewed vectors as representative of 307:. Working in a Euclidean plane, he made equipollent any pair of 13225: 12041: 11923: 11845: 11017: 9262: 5677: 4442: 3825: 1691: 1687: 1679: 764:, more precisely, a Euclidean vector. The equivalence class of 288: 11514: 7533: 6376:

for an explanation of cosine). Geometrically, this means that

3580:

to form the given vector. Typically, these components are the

11926:, or more generally any cross product of two (true) vectors. 11914:

of this angular velocity vector points to the right, but the

11907: 11826: 11433: 11389:

is a vector with dimensions of mass×length/time (N m s ) and

11386: 11279: 6361: 2233:), vectors are identified with triples of scalar components: 2220:

is usually deemed not necessary (and is indeed rarely used).

2111: 1864: 1655: 1297: 1285: 1222: 1154:

of the space, as the coordinates on this basis of the vector

845: 565: 303:

abstracted the basic idea when he established the concept of

279:

used to describe it. Other vector-like objects that describe

264: 12515:"U. Guelph Physics Dept., "Torque and Angular Acceleration"" 10816:. If it represents, for example, a force, the "scale" is of 6384:

are drawn with a common start point, and then the length of

5940:

a vector. A unit vector is often indicated with a hat as in

2083: 1296:-axis as 'up'). Another quantity represented by a vector is 855:

In modern geometry, Euclidean spaces are often defined from

439:. Clifford simplified the quaternion study by isolating the 12813:

Vectors, Tensors and the Basic Equations of Fluid Mechanics

6746:

can be interpreted as the area of the parallelogram having

5318:

at the same point, and then draw an arrow from the head of

2068:

head on and viewing the flights of an arrow from the back.

1589: 2041:. The length of the arrow is proportional to the vector's 1557:{\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.} 552:; such a condition may be emphasized calling the result a 12185:"Earliest Known Uses of Some of the Words of Mathematics" 11903: 11278:

of a point or particle is a vector, its length gives the

426: 421:

carried the quaternion standard after Hamilton. His 1867

702:), vectors were introduced (during the 19th century) as 12685:

Introduction to Tensor Calculus and Continuum Mechanics

2118:

of the endpoint of the vector, with respect to a given

1867:(~) or a wavy underline drawn beneath the symbol, e.g. 1863:, especially in handwriting. Alternatively, some use a 10874: 10801:

Vectors have many uses in physics and other sciences.

9190:

This matrix equation relates the scalar components of

9147: 9021: 8985: 8760:

Replacing each dot product with a unique scalar gives

8076:

and subtraction can be performed. One way to express

7192: 3259:{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } 2527: 2132:) of the vector on the axes of the coordinate system. 1588:, which supplies an algebraic characterization of the 1228:

of its Cartesian coordinates, and every vector to its

675:. If the Euclidean space is equipped with a choice of 656:{\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}} 144: 96:. Euclidean vectors can be added and scaled to form a 12359:

London, Edinburgh & Dublin Philosophical Magazine

12146:= "I carry". For historical development of the word 11753: 11445: 11399: 11292: 11083: 10914: 10904:) in three-dimensional space can be represented as a 10574: 10112: 9883: 9277: 8979: 8768: 8404: 8112: 7919: 7835: 7666: 7579: 7331: 7156: 7042: 6763: 6556: 6407: 6289: 6188: 5987: 5857: 5730: 5430: 5141: 4826: 4681: 4572: 4463: 4348: 4243: 4138: 4023: 3876: 3742: 3679: 3612: 3415: 3334: 3272: 3234: 3006: 2829: 2657: 2513: 2391: 2309: 2239: 2194: 2149: 1972: 1922: 1873: 1834: 1798: 1775: 1752: 1432: 1360: 1195: 1160: 1104: 1056: 1024: 991: 953: 911: 873: 782: 622: 585: 12830: 12592:

Applied mathematics in integrated navigation systems

9844:

The properties of a direction cosine matrix, C are:

5975:, scale the vector by the reciprocal of its length ‖ 3606:

Moreover, the use of Cartesian unit vectors such as

609:{\displaystyle {\stackrel {\,\longrightarrow }{AB}}} 12151: 11641:

may be too technical for most readers to understand

10847:represents the position vector of a particle, then 5845:This happens to be equal to the square root of the 2094:-dimensional Euclidean space can be represented as 947:for details of this construction). The elements of 12781: 12660: 12152: 11768: 11504: 11422: 11342: 11244: 11008: 10773: 10547: 10098: 9716: 9180: 8961: 8750: 8386: 8037: 7897: 7784: 7644: 7523: 7315: 7105: 6984: 6622: 6495: 6388:is multiplied with the length of the component of 6346: 6203: 6153: 5896: 5807: 5546: 5292: 4977: 4774: 4663: 4554: 4432: 4334: 4229: 4117: 4005: 3787: 3724: 3657: 3531: 3347: 3317: 3258: 3143: 2990: 2789: 2633: 2485: 2365: 2295: 2212: 2178: 1982: 1945:{\displaystyle {\stackrel {\longrightarrow }{AB}}} 1944: 1892: 1849: 1806: 1783: 1760: 1556: 1378: 1213: 1181: 1122: 1071: 1039: 1006: 966: 927: 889: 803: 655: 608: 169: 12547:Handbook of mathematics and computational science 5994: 3779: 3764: 3749: 3716: 3701: 3686: 3649: 3634: 3619: 3339: 3309: 3294: 3279: 2366:{\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).} 2296:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).} 1997:), as illustrated in the figure. Here, the point 13343: 12834:; Leighton, R.; Sands, M. (2005). "Chapter 11". 12492:may be resolved into two components parallel to 12207:(2nd. ed.). London: Clarendon Press. 2001. 7140:with respect to a right-handed orthonormal basis 179:A vector is what is needed to "carry" the point 170:{\textstyle {\stackrel {\longrightarrow }{AB}}.} 12427: 12425: 1566: 828:) and same direction (e.g., the direction from 240:. These operations and associated laws qualify 12840:. Vol. I (2nd ed.). Addison Wesley. 11423:{\displaystyle {\mathbf {F} }=m{\mathbf {a} }} 5849:, discussed below, of the vector with itself: 3561: 2813:, respectively. In terms of these, any vector 863:is defined as a set to which is associated an 568:applied to a body has a point of contact (see 515:is defined more generally as any element of a 27:Geometric object that has length and direction 12915: 12543: 11378:of velocity. Its dimensions are length/time. 11365:of position. Its dimensions are length/time. 11282:. For constant velocity the position at time 11268:. Displacement has the dimensions of length. 6719:also becomes a right-handed system (although 5613:over vector addition in the following sense: 5042:This addition method is sometimes called the 5008:, and then drawing an arrow from the tail of 4985:The resulting vector is sometimes called the 4443:Opposite, parallel, and antiparallel vectors 3577: 1275: 425:included extensive treatment of the nabla or 12745:Kane, Thomas R.; Levinson, David A. (1996), 12744: 12577: 12450: 12422: 9833:" is the transpose of "rotation matrix from 9825:. This means that the "rotation matrix from 6668:unit vectors that are perpendicular to both 5016:. The new arrow drawn represents the vector 3855: 1627:(which is important to our understanding of 1235: 11594:. Unsourced material may be challenged and 11553:Learn how and when to remove these messages 11515:Vectors, pseudovectors, and transformations 7534:Conversion between multiple Cartesian bases 4801: 1674:. If you change units (a special case of a 12922: 12908: 12681: 12625: 12573: 12571: 12569: 12567: 12112: 11946:, which distinguishes between vectors and 10828: 8396:Distributing the dot-multiplication gives 544:A Euclidean vector may possess a definite 11687:Learn how and when to remove this message 11669:Learn how and when to remove this message 11653:, without removing the technical details. 11614:Learn how and when to remove this message 6614: 4771: 4429: 3813:to a surface (see figure). Moreover, the 3407:(note the difference in boldface). Thus, 1893:{\displaystyle {\underset {^{\sim }}{a}}} 1814:, or in lowercase italic boldface, as in 1550: 1198: 1107: 1059: 1027: 994: 645: 598: 453:, who was exposed to quaternions through 30:For mathematical vectors in general, see 12749:, Sunnyvale, California: OnLine Dynamics 12544:Harris, John W.; Stöcker, Horst (1998). 12480:coplanar with two non-collinear vectors 12415:, and its equivalence class is called a 11973:Covariance and contravariance of vectors 11883:. Ordinary vectors are sometimes called 11375: 11362: 7001: 6686: 6269:, or, since its result is a scalar, the 5910: 5705:|, which is not to be confused with the 5589: 5386: 5376: 3594: 2495:These numbers are often arranged into a 2070: 1916:(see figure), it can also be denoted as 1824:letters are typically used to represent 1684:covariance and contravariance of vectors 840:, which have no direction. For example, 760:. Such an equivalence class is called a 40: 12655: 12631: 12564: 12539: 12537: 12535: 12313: 12311: 12309: 12307: 12182: 8100:is to use column matrices along with a 7819:} that is not necessarily aligned with 5395:A vector may also be multiplied, or re- 5326:. This new arrow represents the vector 3776: 3761: 3698: 3683: 2055:On a two-dimensional diagram, a vector 1182:{\displaystyle {\overrightarrow {OP}}.} 804:{\displaystyle {\overrightarrow {AB}}.} 14: 13344: 13313:Comparison of linear algebra libraries 12803: 12725: 12705:Encyclopedic Dictionary of Mathematics 12589: 12101: 11381: 11016:The position vector has dimensions of 9851:the inverse is equal to the transpose; 5560:stretches a vector out by a factor of 5410:, these real numbers are often called 2213:{\displaystyle {\overrightarrow {OA}}} 1379:{\displaystyle {\overrightarrow {AB}}} 1316:. Other physical vectors, such as the 928:{\displaystyle {\overrightarrow {E}},} 890:{\displaystyle {\overrightarrow {E}},} 689: 12903: 12776: 12457:Thermodynamics and Differential Forms 12285: 12127: 11651:make it understandable to non-experts 10565:. In two dimensions this is simply a 9776:basis, the matrix containing all the 6392:that points in the same direction as 5556:Intuitively, multiplying by a scalar 2643:Another way to represent a vector in 1331: 967:{\displaystyle {\overrightarrow {E}}} 459:Treatise on Electricity and Magnetism 362:of equipollent directed segments. As 12809: 12754: 12532: 12387: 12304: 12257: 12255: 12253: 12251: 12226: 12224: 12177:participating institution membership 11625: 11592:adding citations to reliable sources 11559: 11518: 6757:The cross product can be written as 6691:An illustration of the cross product 6636:is the measure of the angle between 6540:, is a vector perpendicular to both 4451:if they have the same magnitude but 3573: 1354:in space determine the bound vector 859:. More precisely, a Euclidean space 12702: 12370: 12123: 11906:, and looking forward, each of the 10875:Position, velocity and acceleration 10804: 9872: 5082:. One can check geometrically that 2179:{\displaystyle \mathbf {a} =(2,3).} 1975: 867:of finite dimension over the reals 24: 12929: 11739:must be similarly transformed via 11374:of a point is vector which is the 9848:the determinant is unity, |C| = 1; 5360: 5026: 3348:{\displaystyle \mathbf {\hat {}} } 2623: 2082: 2047: 1726: 1716: 1649: 423:Elementary Treatise of Quaternions 251:Vectors play an important role in 25: 13398: 13377:Vectors (mathematics and physics) 12855: 12550:. Birkhäuser. Chapter 6, p. 332. 12248: 12232:"vector | Definition & Facts" 12221: 11898:One example of a pseudovector is 11534:This section has multiple issues. 7907:and the scalar components in the 7150:having the three vectors as rows 6998:instead of a vector (see below). 6528:) is only meaningful in three or 4820:of two vectors may be defined as 3862:Vector notation § Operations 2503:, particularly when dealing with 1256:is also directed rightward, then 1240:Since the physicist's concept of 1214:{\displaystyle \mathbb {R} ^{n},} 1123:{\displaystyle \mathbb {R} ^{n}.} 374:, Hamilton considered the vector 84:) is a geometric object that has 13326: 13325: 13303:Basic Linear Algebra Subprograms 13061: 12784:Geometry: A comprehensive course 12036:Tangential and normal components 11630: 11564: 11523: 11485: 11468: 11454: 11415: 11402: 11326: 11315: 11296: 11252:which specifies the position of 11228: 11182: 11136: 11096: 11086: 10992: 10965: 10938: 10917: 10758: 10746: 10678: 10651: 10618: 10591: 10527: 10481: 10431: 10385: 10332: 10305: 10278: 10251: 10214: 10187: 10160: 10133: 10082: 10036: 9987: 9960: 9927: 9900: 9700: 9685: 9652: 9637: 9604: 9589: 9556: 9541: 9508: 9493: 9460: 9445: 9412: 9397: 9364: 9349: 9316: 9301: 9266: 8731: 8716: 8698: 8683: 8665: 8650: 8618: 8603: 8585: 8570: 8552: 8537: 8505: 8490: 8472: 8457: 8439: 8424: 8367: 8349: 8331: 8313: 8278: 8260: 8242: 8224: 8189: 8171: 8153: 8135: 8018: 8009: 7981: 7972: 7944: 7935: 7885: 7867: 7849: 7837: 7765: 7756: 7728: 7719: 7691: 7682: 7629: 7611: 7593: 7581: 7571:is expressed, by definition, as 7511: 7503: 7495: 7478: 7470: 7462: 7445: 7437: 7429: 7412: 7404: 7396: 7382: 7374: 7366: 7352: 7344: 7336: 7177: 7169: 7161: 7093: 7085: 7074: 7063: 7055: 7047: 6968: 6902: 6836: 6776: 6766: 6616: 6591: 6578: 6566: 6558: 6505: 6417: 6409: 6324: 6311: 6299: 6291: 6141: 6129: 6099: 6087: 6057: 6045: 6015: 6006: 5991: 5885: 5877: 5863: 5736: 5531: 5497: 5463: 5435: 5277: 5233: 5189: 5151: 5143: 4962: 4918: 4874: 4836: 4828: 4650: 4623: 4596: 4575: 4541: 4514: 4487: 4466: 4321: 4294: 4267: 4246: 4216: 4189: 4162: 4141: 4101: 4074: 4047: 4026: 3968: 3924: 3880: 3837:to the radius and the latter is 3746: 3713: 3646: 3631: 3616: 3521: 3501: 3481: 3456: 3441: 3426: 3417: 3306: 3291: 3276: 3252: 3244: 3236: 3127: 3100: 3073: 3047: 3032: 3017: 3008: 2831: 2749: 2705: 2661: 2647:-dimensions is to introduce the 2515: 2393: 2311: 2241: 2151: 1800: 1777: 1754: 1072:{\displaystyle \mathbb {R} ^{n}} 1040:{\displaystyle \mathbb {R} ^{n}} 1007:{\displaystyle \mathbb {R} ^{n}} 738:being equipollent if the points 32:Vector (mathematics and physics) 13201:Seven-dimensional cross product 12837:The Feynman Lectures on Physics 12583: 12507: 12461: 12393: 12376: 12364: 12351: 12080: 11542:or discuss these issues on the 10908:whose base point is the origin 10785:seven-dimensional cross product 5365:The subtraction of two vectors 4741: 4711: 4402: 4375: 3866:The following section uses the 2122:, and are typically called the 1983:{\displaystyle {\mathfrak {a}}} 1742:Vectors are usually denoted in 1701:, a vector is any element of a 12760:Introduction to Linear Algebra 12279: 12197: 12132: 12117: 12106: 12095: 11735:, then a contravariant vector 11718:, so that a coordinate vector 11496: 11462: 11222: 11196: 11176: 11150: 11130: 11104: 10741: 10695: 10689: 10635: 10629: 10575: 10521: 10495: 10475: 10449: 10425: 10399: 10379: 10353: 10343: 10235: 10225: 10117: 10076: 10050: 10030: 10004: 9998: 9944: 9938: 9884: 8359: 8305: 8270: 8216: 8181: 8127: 7515: 7491: 7482: 7458: 7449: 7425: 7416: 7392: 7386: 7362: 7356: 7332: 7181: 7157: 7097: 7081: 7067: 7043: 6962: 6916: 6896: 6850: 6830: 6784: 6611: 6605: 6595: 6587: 6582: 6574: 6328: 6320: 6315: 6307: 6240: 6195: 6163: 6133: 6125: 6091: 6083: 6049: 6041: 6019: 6011: 5915:The normalization of a vector 5906: 5867: 5859: 5817:which is a consequence of the 5740: 5732: 5526: 5510: 5492: 5476: 5458: 5442: 5272: 5246: 5228: 5202: 5184: 5158: 4957: 4931: 4913: 4887: 4869: 4843: 4000: 3982: 3956: 3938: 3912: 3894: 2979: 2961: 2945: 2927: 2911: 2893: 2877: 2838: 2823:can be expressed in the form: 2781: 2763: 2737: 2719: 2693: 2675: 2619: 2579: 2477: 2400: 2357: 2318: 2287: 2248: 2170: 2158: 2143:= (2, 3) is simply written as 1935: 1841: 1709:and is often represented as a 1547: 1526: 1520: 1484: 1478: 1457: 1451: 1433: 646: 599: 499:. It is formally defined as a 157: 13: 1: 12620: 12205:The Oxford English Dictionary 11393:is the scalar multiplication 9726:By referring collectively to 7794:In another orthonormal basis 7654:The scalar components in the 6182:, and it is commonly denoted 5842:are orthogonal unit vectors. 3671:cylindrical coordinate system 1828:.) Other conventions include 1288:. For instance, the velocity 13043:Eigenvalues and eigenvectors 12399:In some old texts, the pair 12322:A History of Vector Analysis 5594:The scalar multiplications − 5066:is one of the diagonals. If 5031:The addition of two vectors 3833:of an object. The former is 1807:{\displaystyle \mathbf {w} } 1784:{\displaystyle \mathbf {v} } 1761:{\displaystyle \mathbf {u} } 1579:between two vectors. If the 1567:Euclidean and affine vectors 1284:, the magnitude of which is 1221:by mapping any point to the 852:are represented by vectors. 7: 12892:A conceptual introduction ( 12868:Encyclopedia of Mathematics 12733:Encyclopedia of Mathematics 11936: 11432:Work is the dot product of 5408:conventional vector algebra 4127: 3868:Cartesian coordinate system 3734:spherical coordinate system 3562:Decomposition or resolution 2811:Cartesian coordinate system 2375:This can be generalised to 2120:Cartesian coordinate system 2100:Cartesian coordinate system 1731:Vector arrow pointing from 1338:Cartesian coordinate system 478: 463:Elements of Vector Analysis 263:of a moving object and the 10: 13403: 12762:(2nd ed.). Springer. 12682:Heinbockel, J. H. (2001), 12590:Rogers, Robert M. (2007). 10832: 10794: 10790: 10106:and in four dimensions as 7911:basis are, by definition, 7658:basis are, by definition, 7005: 6509: 6244: 6204:{\displaystyle {\vec {0}}} 6167: 5925: 5568:is an integer) as placing 5422:. The resulting vector is 5380: 4805: 3859: 3823:of a vector relate to the 3565: 1850:{\displaystyle {\vec {a}}} 1720: 1290:5 meters per second upward 1276:In physics and engineering 294: 29: 13321: 13283: 13239: 13176: 13128: 13070: 13059: 12955: 12937: 12164:Oxford English Dictionary 11895:of two ordinary vectors. 11701:coordinate transformation 5716:can be computed with the 5712:The length of the vector 5670: 5651:. One can also show that 5609:Scalar multiplication is 5004:at the head of the arrow 3856:Properties and operations 3225:(or scalar projections). 2114:). These numbers are the 1236:Examples in one dimension 415:Theorie der Ebbe und Flut 12880:Online vector identities 12578:Kane & Levinson 1996 12073: 11357:is the position at time 10879:The position of a point 10823:proportionality constant 10561:does, whose result is a 9226:). Each matrix element 5821:since the basis vectors 5024:, as illustrated below: 4802:Addition and subtraction 3850:inertial reference frame 3590:resolved with respect to 2079:with coordinates (2, 3). 1386:pointing from the point 1084:standard Euclidean space 756:, in this order, form a 437:William Kingdon Clifford 413:. Grassmann's 1840 work 12688:, Trafford Publishing, 12626:Mathematical treatments 12236:Encyclopedia Britannica 12169:Oxford University Press 12142:of vehere, "to carry"/ 10829:Vector-valued functions 8102:direction cosine matrix 5338:being the opposite of 3550:is compatible with the 1638:(for free vectors) and 1304:, linear acceleration, 1142:, one may also find an 1098:to the Euclidean space 45:A vector pointing from 36:Vector (disambiguation) 34:. For other uses, see 13028:Row and column vectors 12726:Ivanov, A.B. (2001) , 12437:Mathematics LibreTexts 12357:W. R. Hamilton (1846) 12067:Vector-valued function 11770: 11506: 11424: 11344: 11246: 11010: 10835:Vector-valued function 10775: 10549: 10100: 9718: 9182: 8963: 8752: 8388: 8039: 7899: 7786: 7646: 7525: 7317: 7107: 6986: 6692: 6624: 6497: 6374:trigonometric function 6360:is the measure of the 6348: 6263:(sometimes called the 6205: 6155: 5947:To normalize a vector 5923: 5898: 5809: 5701:‖ or, less commonly, | 5606: 5587:= 2) are given below: 5548: 5392: 5373: 5294: 5039: 4979: 4776: 4665: 4556: 4434: 4336: 4231: 4119: 4007: 3848:coordinate system, or 3789: 3726: 3659: 3600: 3568:Basis (linear algebra) 3533: 3349: 3319: 3260: 3145: 2992: 2791: 2635: 2487: 2367: 2297: 2214: 2180: 2139:= (0, 0) to the point 2087: 2080: 2052: 1984: 1946: 1894: 1851: 1808: 1785: 1762: 1739: 1621:pseudo-Euclidean space 1558: 1380: 1215: 1183: 1124: 1079:is often presented as 1073: 1041: 1008: 968: 929: 891: 805: 657: 610: 391: 354:) and a 3-dimensional 321:William Rowan Hamilton 171: 53: 13033:Row and column spaces 12978:Scalar multiplication 12292:mathworld.wolfram.com 11838:differential geometry 11771: 11769:{\displaystyle ^{-1}} 11507: 11425: 11361:= 0. Velocity is the 11345: 11247: 11011: 10776: 10550: 10101: 9787:transformation matrix 9719: 9183: 8964: 8753: 8389: 8040: 7900: 7787: 7647: 7526: 7318: 7108: 7014:scalar triple product 7008:Scalar triple product 7002:Scalar triple product 6987: 6690: 6625: 6498: 6349: 6206: 6156: 5914: 5899: 5810: 5593: 5549: 5420:scalar multiplication 5390: 5383:Scalar multiplication 5377:Scalar multiplication 5364: 5310:, place the tails of 5295: 5030: 4980: 4806:Further information: 4777: 4666: 4557: 4435: 4337: 4232: 4120: 4008: 3820:tangential components 3790: 3727: 3660: 3598: 3566:Further information: 3534: 3350: 3320: 3261: 3146: 2993: 2792: 2636: 2488: 2368: 2298: 2215: 2181: 2086: 2074: 2051: 1985: 1947: 1895: 1852: 1809: 1786: 1763: 1730: 1723:Vector representation 1721:Further information: 1559: 1381: 1216: 1184: 1148:Cartesian coordinates 1125: 1074: 1042: 1009: 984:real coordinate space 969: 930: 892: 806: 714:of points; two pairs 667:if the quadrilateral 658: 611: 501:directed line segment 407:Comte de Saint-Venant 386: 172: 121:directed line segment 44: 13168:Gram–Schmidt process 13120:Gaussian elimination 12703:Itô, Kiyosi (1993), 11842:contravariant vector 11751: 11705:contravariant vector 11588:improve this section 11443: 11397: 11290: 11081: 10912: 10572: 10110: 9881: 9275: 9210:) with those in the 8977: 8766: 8402: 8110: 7917: 7833: 7664: 7577: 7329: 7154: 7040: 7022:mixed triple product 6761: 6727:are not necessarily 6554: 6405: 6287: 6186: 5985: 5855: 5728: 5428: 5406:. In the context of 5342:, see drawing. And 5139: 5054:form the sides of a 4824: 4679: 4570: 4461: 4346: 4241: 4136: 4021: 3874: 3740: 3677: 3610: 3556:summation convention 3413: 3332: 3270: 3232: 3200:(see figure), while 3004: 2827: 2655: 2511: 2389: 2379:Euclidean space (or 2307: 2237: 2227:Euclidean space (or 2192: 2147: 1970: 1920: 1871: 1832: 1796: 1773: 1750: 1430: 1358: 1306:angular acceleration 1193: 1158: 1140:Gram–Schmidt process 1102: 1054: 1050:The Euclidean space 1022: 989: 951: 909: 871: 780: 620: 583: 467:Edwin Bidwell Wilson 451:Josiah Willard Gibbs 313:equivalence relation 206:algebraic operations 142: 112:units of measurement 76:(sometimes called a 13372:Concepts in physics 13298:Numerical stability 13178:Multilinear algebra 13153:Inner product space 13003:Linear independence 12894:applied mathematics 12890:Introducing Vectors 12804:Physical treatments 12339:on January 26, 2004 12286:Weisstein, Eric W. 12167:(Online ed.). 12021:Position (geometry) 11836:In the language of 11391:Newton's second law 11382:Force, energy, work 10843:. For instance, if 6703:is defined so that 6281:and is defined as: 5919:into a unit vector 5819:Pythagorean theorem 5799: 5781: 5763: 5709:(a scalar "norm"). 4017:will be written as 3870:with basis vectors 3221:are the respective 1397:-axis to the point 865:inner product space 704:equivalence classes 690:Further information 455:James Clerk Maxwell 432:Elements of Dynamic 281:physical quantities 106:is a vector-valued 13387:Euclidean geometry 13008:Linear combination 12382:Formerly known as 12267:www.mathsisfun.com 12140:perfect participle 11766: 11722:is transformed to 11502: 11420: 11340: 11242: 11006: 10818:physical dimension 10771: 10545: 10543: 10096: 9714: 9712: 9178: 9169: 9136: 9007: 8959: 8957: 8748: 8746: 8384: 8382: 8035: 8033: 7895: 7782: 7780: 7642: 7521: 7313: 7307: 7135:are right-handed. 7121:linearly dependent 7103: 7034:) and defined as: 6982: 6695:The cross product 6693: 6660:which completes a 6620: 6548:and is defined as 6493: 6344: 6201: 6151: 5924: 5894: 5805: 5785: 5767: 5749: 5607: 5544: 5393: 5374: 5290: 5125:The difference of 5044:parallelogram rule 5040: 4975: 4772: 4661: 4552: 4453:opposite direction 4430: 4332: 4227: 4115: 4003: 3785: 3722: 3655: 3601: 3529: 3355:typically denotes 3345: 3315: 3256: 3182:vector projections 3141: 2988: 2787: 2631: 2570: 2483: 2363: 2293: 2210: 2176: 2130:scalar projections 2096:coordinate vectors 2088: 2081: 2053: 1980: 1942: 1890: 1888: 1847: 1804: 1781: 1758: 1740: 1629:special relativity 1554: 1376: 1332:In Cartesian space 1252:. If the positive 1211: 1179: 1120: 1069: 1037: 1014:equipped with the 1004: 964: 925: 887: 801: 700:synthetic geometry 696:Euclidean geometry 653: 606: 497:relative direction 419:Peter Guthrie Tait 370:to complement the 319:was introduced by 167: 118:, formulated as a 54: 13382:Analytic geometry 13339: 13338: 13206:Geometric algebra 13163:Kronecker product 12998:Linear projection 12983:Vector projection 12847:978-0-8053-9046-9 12823:978-0-486-66110-0 12810:Aris, R. (1990). 12718:978-0-262-59020-4 12674:978-0-471-00007-5 12648:978-0-471-00005-1 12175:(Subscription or 11968:Coordinate system 11713:invertible matrix 11697: 11696: 11689: 11679: 11678: 11671: 11624: 11623: 11616: 11557: 11023:Given two points 9785:is known as the " 7509: 7501: 7476: 7468: 7443: 7435: 7410: 7402: 7380: 7372: 7350: 7342: 7175: 7167: 7061: 7053: 7016:(also called the 6648:is a unit vector 6520:(also called the 6198: 6137: 6095: 6053: 6023: 5997: 5889: 5800: 4455:; so two vectors 3964: 3920: 3782: 3767: 3752: 3719: 3704: 3689: 3652: 3637: 3622: 3342: 3338: 3312: 3297: 3282: 3223:scalar components 3177:vector components 2987: 2745: 2701: 2607: 2594: 2225:three dimensional 2208: 2125:scalar components 1939: 1875: 1844: 1711:coordinate vector 1374: 1230:coordinate vector 1174: 1144:orthonormal basis 962: 920: 882: 796: 776:is often denoted 650: 603: 503:, or arrow, in a 435:was published by 399:Hermann Grassmann 384:of a quaternion: 327:, which is a sum 301:Giusto Bellavitis 277:coordinate system 187:; the Latin word 161: 138:, and denoted by 108:physical quantity 18:Euclidean vectors 16:(Redirected from 13394: 13357:Abstract algebra 13329: 13328: 13211:Exterior algebra 13148:Hadamard product 13065: 13053:Linear equations 12924: 12917: 12910: 12901: 12900: 12876: 12851: 12832:Feynman, Richard 12827: 12799: 12787: 12773: 12750: 12740: 12721: 12707:(2nd ed.), 12698: 12678: 12666: 12652: 12614: 12613: 12587: 12581: 12580:, pp. 20–22 12575: 12562: 12561: 12541: 12530: 12529: 12527: 12526: 12517:. Archived from 12511: 12505: 12503: 12497: 12491: 12485: 12479: 12465: 12459: 12454: 12448: 12447: 12445: 12444: 12429: 12420: 12410: 12397: 12391: 12380: 12374: 12368: 12362: 12361:3rd series 29 27 12355: 12349: 12347: 12345: 12344: 12338: 12332:. Archived from 12331: 12318:Michael J. Crowe 12315: 12302: 12301: 12299: 12298: 12283: 12277: 12276: 12274: 12273: 12259: 12246: 12245: 12243: 12242: 12228: 12219: 12218: 12201: 12195: 12194: 12192: 12191: 12180: 12172: 12160: 12136: 12130: 12126:, p. 1678; 12121: 12115: 12110: 12104: 12099: 12087: 12084: 12016:Parity (physics) 11996:Ausdehnungslehre 11958:Clifford algebra 11900:angular velocity 11784:consists of the 11779: 11775: 11773: 11772: 11767: 11765: 11764: 11734: 11692: 11685: 11674: 11667: 11663: 11660: 11654: 11634: 11633: 11626: 11619: 11612: 11608: 11605: 11599: 11568: 11560: 11549: 11527: 11526: 11519: 11511: 11509: 11508: 11503: 11495: 11494: 11489: 11488: 11478: 11477: 11472: 11471: 11458: 11457: 11429: 11427: 11426: 11421: 11419: 11418: 11406: 11405: 11349: 11347: 11346: 11341: 11336: 11335: 11330: 11329: 11319: 11318: 11306: 11305: 11300: 11299: 11251: 11249: 11248: 11243: 11238: 11237: 11232: 11231: 11221: 11220: 11208: 11207: 11192: 11191: 11186: 11185: 11175: 11174: 11162: 11161: 11146: 11145: 11140: 11139: 11129: 11128: 11116: 11115: 11100: 11099: 11090: 11089: 11015: 11013: 11012: 11007: 11002: 11001: 10996: 10995: 10988: 10987: 10975: 10974: 10969: 10968: 10961: 10960: 10948: 10947: 10942: 10941: 10934: 10933: 10921: 10920: 10805:Length and units 10780: 10778: 10777: 10772: 10767: 10766: 10761: 10755: 10754: 10749: 10740: 10739: 10730: 10729: 10717: 10716: 10707: 10706: 10688: 10687: 10682: 10681: 10674: 10673: 10661: 10660: 10655: 10654: 10647: 10646: 10628: 10627: 10622: 10621: 10614: 10613: 10601: 10600: 10595: 10594: 10587: 10586: 10559:exterior product 10554: 10552: 10551: 10546: 10544: 10537: 10536: 10531: 10530: 10520: 10519: 10507: 10506: 10491: 10490: 10485: 10484: 10474: 10473: 10461: 10460: 10441: 10440: 10435: 10434: 10424: 10423: 10411: 10410: 10395: 10394: 10389: 10388: 10378: 10377: 10365: 10364: 10342: 10341: 10336: 10335: 10328: 10327: 10315: 10314: 10309: 10308: 10301: 10300: 10288: 10287: 10282: 10281: 10274: 10273: 10261: 10260: 10255: 10254: 10247: 10246: 10224: 10223: 10218: 10217: 10210: 10209: 10197: 10196: 10191: 10190: 10183: 10182: 10170: 10169: 10164: 10163: 10156: 10155: 10143: 10142: 10137: 10136: 10129: 10128: 10105: 10103: 10102: 10097: 10092: 10091: 10086: 10085: 10075: 10074: 10062: 10061: 10046: 10045: 10040: 10039: 10029: 10028: 10016: 10015: 9997: 9996: 9991: 9990: 9983: 9982: 9970: 9969: 9964: 9963: 9956: 9955: 9937: 9936: 9931: 9930: 9923: 9922: 9910: 9909: 9904: 9903: 9896: 9895: 9873:Other dimensions 9821:is equal to its 9723: 9721: 9720: 9715: 9713: 9709: 9708: 9703: 9694: 9693: 9688: 9675: 9674: 9661: 9660: 9655: 9646: 9645: 9640: 9627: 9626: 9613: 9612: 9607: 9598: 9597: 9592: 9579: 9578: 9565: 9564: 9559: 9550: 9549: 9544: 9531: 9530: 9517: 9516: 9511: 9502: 9501: 9496: 9483: 9482: 9469: 9468: 9463: 9454: 9453: 9448: 9435: 9434: 9421: 9420: 9415: 9406: 9405: 9400: 9387: 9386: 9373: 9372: 9367: 9358: 9357: 9352: 9339: 9338: 9325: 9324: 9319: 9310: 9309: 9304: 9291: 9290: 9259:direction cosine 9237:direction cosine 9187: 9185: 9184: 9179: 9174: 9173: 9141: 9140: 9133: 9132: 9121: 9120: 9109: 9108: 9095: 9094: 9083: 9082: 9071: 9070: 9057: 9056: 9045: 9044: 9033: 9032: 9012: 9011: 8968: 8966: 8965: 8960: 8958: 8948: 8947: 8932: 8931: 8916: 8915: 8886: 8885: 8870: 8869: 8854: 8853: 8824: 8823: 8808: 8807: 8792: 8791: 8757: 8755: 8754: 8749: 8747: 8740: 8739: 8734: 8725: 8724: 8719: 8707: 8706: 8701: 8692: 8691: 8686: 8674: 8673: 8668: 8659: 8658: 8653: 8627: 8626: 8621: 8612: 8611: 8606: 8594: 8593: 8588: 8579: 8578: 8573: 8561: 8560: 8555: 8546: 8545: 8540: 8514: 8513: 8508: 8499: 8498: 8493: 8481: 8480: 8475: 8466: 8465: 8460: 8448: 8447: 8442: 8433: 8432: 8427: 8393: 8391: 8390: 8385: 8383: 8376: 8375: 8370: 8358: 8357: 8352: 8340: 8339: 8334: 8322: 8321: 8316: 8287: 8286: 8281: 8269: 8268: 8263: 8251: 8250: 8245: 8233: 8232: 8227: 8198: 8197: 8192: 8180: 8179: 8174: 8162: 8161: 8156: 8144: 8143: 8138: 8044: 8042: 8041: 8036: 8034: 8027: 8026: 8021: 8012: 7990: 7989: 7984: 7975: 7953: 7952: 7947: 7938: 7904: 7902: 7901: 7896: 7894: 7893: 7888: 7876: 7875: 7870: 7858: 7857: 7852: 7840: 7827:is expressed as 7791: 7789: 7788: 7783: 7781: 7774: 7773: 7768: 7759: 7737: 7736: 7731: 7722: 7700: 7699: 7694: 7685: 7651: 7649: 7648: 7643: 7638: 7637: 7632: 7620: 7619: 7614: 7602: 7601: 7596: 7584: 7567:basis, a vector 7530: 7528: 7527: 7522: 7514: 7507: 7506: 7499: 7498: 7481: 7474: 7473: 7466: 7465: 7448: 7441: 7440: 7433: 7432: 7415: 7408: 7407: 7400: 7399: 7385: 7378: 7377: 7370: 7369: 7355: 7348: 7347: 7340: 7339: 7322: 7320: 7319: 7314: 7312: 7311: 7304: 7303: 7292: 7291: 7280: 7279: 7266: 7265: 7254: 7253: 7242: 7241: 7228: 7227: 7216: 7215: 7204: 7203: 7180: 7173: 7172: 7165: 7164: 7112: 7110: 7109: 7104: 7096: 7088: 7077: 7066: 7059: 7058: 7051: 7050: 6991: 6989: 6988: 6983: 6978: 6977: 6972: 6971: 6961: 6960: 6951: 6950: 6938: 6937: 6928: 6927: 6912: 6911: 6906: 6905: 6895: 6894: 6885: 6884: 6872: 6871: 6862: 6861: 6846: 6845: 6840: 6839: 6829: 6828: 6819: 6818: 6806: 6805: 6796: 6795: 6780: 6779: 6770: 6769: 6629: 6627: 6626: 6621: 6619: 6598: 6594: 6585: 6581: 6569: 6561: 6502: 6500: 6499: 6494: 6489: 6488: 6479: 6478: 6466: 6465: 6456: 6455: 6443: 6442: 6433: 6432: 6420: 6412: 6353: 6351: 6350: 6345: 6331: 6327: 6318: 6314: 6302: 6294: 6273:) is denoted by 6236: 6210: 6208: 6207: 6202: 6200: 6199: 6191: 6181: 6160: 6158: 6157: 6152: 6150: 6149: 6144: 6138: 6136: 6132: 6123: 6122: 6113: 6108: 6107: 6102: 6096: 6094: 6090: 6081: 6080: 6071: 6066: 6065: 6060: 6054: 6052: 6048: 6039: 6038: 6029: 6024: 6022: 6018: 6009: 6004: 5999: 5998: 5990: 5974: 5903: 5901: 5900: 5895: 5890: 5888: 5880: 5875: 5870: 5866: 5814: 5812: 5811: 5806: 5801: 5798: 5793: 5780: 5775: 5762: 5757: 5748: 5743: 5739: 5647:and all scalars 5639:for all vectors 5553: 5551: 5550: 5545: 5540: 5539: 5534: 5525: 5524: 5506: 5505: 5500: 5491: 5490: 5472: 5471: 5466: 5457: 5456: 5438: 5299: 5297: 5296: 5291: 5286: 5285: 5280: 5271: 5270: 5258: 5257: 5242: 5241: 5236: 5227: 5226: 5214: 5213: 5198: 5197: 5192: 5183: 5182: 5170: 5169: 5154: 5146: 4987:resultant vector 4984: 4982: 4981: 4976: 4971: 4970: 4965: 4956: 4955: 4943: 4942: 4927: 4926: 4921: 4912: 4911: 4899: 4898: 4883: 4882: 4877: 4868: 4867: 4855: 4854: 4839: 4831: 4784:Two vectors are 4781: 4779: 4778: 4773: 4767: 4766: 4751: 4750: 4737: 4736: 4721: 4720: 4707: 4706: 4691: 4690: 4673:are opposite if 4670: 4668: 4667: 4662: 4660: 4659: 4654: 4653: 4646: 4645: 4633: 4632: 4627: 4626: 4619: 4618: 4606: 4605: 4600: 4599: 4592: 4591: 4579: 4578: 4561: 4559: 4558: 4553: 4551: 4550: 4545: 4544: 4537: 4536: 4524: 4523: 4518: 4517: 4510: 4509: 4497: 4496: 4491: 4490: 4483: 4482: 4470: 4469: 4447:Two vectors are 4439: 4437: 4436: 4431: 4425: 4424: 4412: 4411: 4398: 4397: 4385: 4384: 4371: 4370: 4358: 4357: 4341: 4339: 4338: 4333: 4331: 4330: 4325: 4324: 4317: 4316: 4304: 4303: 4298: 4297: 4290: 4289: 4277: 4276: 4271: 4270: 4263: 4262: 4250: 4249: 4236: 4234: 4233: 4228: 4226: 4225: 4220: 4219: 4212: 4211: 4199: 4198: 4193: 4192: 4185: 4184: 4172: 4171: 4166: 4165: 4158: 4157: 4145: 4144: 4124: 4122: 4121: 4116: 4111: 4110: 4105: 4104: 4097: 4096: 4084: 4083: 4078: 4077: 4070: 4069: 4057: 4056: 4051: 4050: 4043: 4042: 4030: 4029: 4012: 4010: 4009: 4004: 3978: 3977: 3972: 3971: 3962: 3934: 3933: 3928: 3927: 3918: 3890: 3889: 3884: 3883: 3794: 3792: 3791: 3786: 3784: 3783: 3775: 3769: 3768: 3760: 3754: 3753: 3745: 3731: 3729: 3728: 3723: 3721: 3720: 3712: 3706: 3705: 3697: 3691: 3690: 3682: 3664: 3662: 3661: 3656: 3654: 3653: 3645: 3639: 3638: 3630: 3624: 3623: 3615: 3538: 3536: 3535: 3530: 3525: 3524: 3518: 3517: 3505: 3504: 3498: 3497: 3485: 3484: 3478: 3477: 3465: 3464: 3459: 3450: 3449: 3444: 3435: 3434: 3429: 3420: 3354: 3352: 3351: 3346: 3344: 3343: 3337: 3324: 3322: 3321: 3316: 3314: 3313: 3305: 3299: 3298: 3290: 3284: 3283: 3275: 3265: 3263: 3262: 3257: 3255: 3247: 3239: 3150: 3148: 3147: 3142: 3137: 3136: 3131: 3130: 3123: 3122: 3110: 3109: 3104: 3103: 3096: 3095: 3083: 3082: 3077: 3076: 3069: 3068: 3056: 3055: 3050: 3041: 3040: 3035: 3026: 3025: 3020: 3011: 2997: 2995: 2994: 2989: 2985: 2960: 2959: 2926: 2925: 2892: 2891: 2876: 2875: 2863: 2862: 2850: 2849: 2834: 2822: 2796: 2794: 2793: 2788: 2759: 2758: 2753: 2752: 2743: 2715: 2714: 2709: 2708: 2699: 2671: 2670: 2665: 2664: 2640: 2638: 2637: 2632: 2627: 2626: 2617: 2616: 2605: 2604: 2603: 2592: 2591: 2590: 2575: 2574: 2567: 2566: 2553: 2552: 2539: 2538: 2518: 2492: 2490: 2489: 2484: 2476: 2475: 2463: 2462: 2438: 2437: 2425: 2424: 2412: 2411: 2396: 2384: 2372: 2370: 2369: 2364: 2356: 2355: 2343: 2342: 2330: 2329: 2314: 2302: 2300: 2299: 2294: 2286: 2285: 2273: 2272: 2260: 2259: 2244: 2232: 2219: 2217: 2216: 2211: 2209: 2204: 2196: 2185: 2183: 2182: 2177: 2154: 2017:, and the point 1989: 1987: 1986: 1981: 1979: 1978: 1966:letters such as 1951: 1949: 1948: 1943: 1941: 1940: 1938: 1933: 1925: 1899: 1897: 1896: 1891: 1889: 1887: 1886: 1856: 1854: 1853: 1848: 1846: 1845: 1837: 1813: 1811: 1810: 1805: 1803: 1790: 1788: 1787: 1782: 1780: 1767: 1765: 1764: 1759: 1757: 1746:boldface, as in 1602:exterior product 1596:in space of the 1563: 1561: 1560: 1555: 1417: 1403: 1392: 1385: 1383: 1382: 1377: 1375: 1370: 1362: 1353: 1346: 1314:angular momentum 1225: 1220: 1218: 1217: 1212: 1207: 1206: 1201: 1188: 1186: 1185: 1180: 1175: 1170: 1162: 1153: 1133: 1129: 1127: 1126: 1121: 1116: 1115: 1110: 1093: 1089: 1078: 1076: 1075: 1070: 1068: 1067: 1062: 1046: 1044: 1043: 1038: 1036: 1035: 1030: 1013: 1011: 1010: 1005: 1003: 1002: 997: 973: 971: 970: 965: 963: 955: 934: 932: 931: 926: 921: 913: 896: 894: 893: 888: 883: 875: 862: 835: 831: 827: 810: 808: 807: 802: 797: 792: 784: 775: 755: 737: 725: 662: 660: 659: 654: 652: 651: 649: 643: 642: 634: 625: 615: 613: 612: 607: 605: 604: 602: 596: 588: 509:pure mathematics 379: 349: 340: 176: 174: 173: 168: 163: 162: 160: 155: 147: 78:geometric vector 70:Euclidean vector 21: 13402: 13401: 13397: 13396: 13395: 13393: 13392: 13391: 13362:Vector calculus 13342: 13341: 13340: 13335: 13317: 13279: 13235: 13172: 13124: 13066: 13057: 13023:Change of basis 13013:Multilinear map 12951: 12933: 12928: 12861: 12858: 12848: 12824: 12806: 12796: 12770: 12747:Dynamics Online 12719: 12696: 12675: 12649: 12628: 12623: 12618: 12617: 12602: 12588: 12584: 12576: 12565: 12558: 12542: 12533: 12524: 12522: 12513: 12512: 12508: 12499: 12493: 12487: 12481: 12475: 12466: 12462: 12455: 12451: 12442: 12440: 12431: 12430: 12423: 12400: 12398: 12394: 12381: 12377: 12369: 12365: 12356: 12352: 12348:on the subject. 12342: 12340: 12336: 12329: 12327:"lecture notes" 12325: 12324:; see also his 12316: 12305: 12296: 12294: 12284: 12280: 12271: 12269: 12261: 12260: 12249: 12240: 12238: 12230: 12229: 12222: 12215: 12203: 12202: 12198: 12189: 12187: 12174: 12138:Latin: vectus, 12137: 12133: 12122: 12118: 12113:Heinbockel 2001 12111: 12107: 12100: 12096: 12091: 12090: 12085: 12081: 12076: 12071: 12062:Vector notation 12057:Vector calculus 11939: 11902:. Driving in a 11796:-components of 11757: 11754: 11752: 11749: 11748: 11740: 11723: 11693: 11682: 11681: 11680: 11675: 11664: 11658: 11655: 11647:help improve it 11644: 11635: 11631: 11620: 11609: 11603: 11600: 11585: 11569: 11528: 11524: 11517: 11490: 11484: 11483: 11482: 11473: 11467: 11466: 11465: 11453: 11452: 11444: 11441: 11440: 11414: 11413: 11401: 11400: 11398: 11395: 11394: 11384: 11376:time derivative 11363:time derivative 11356: 11331: 11325: 11324: 11323: 11314: 11313: 11301: 11295: 11294: 11293: 11291: 11288: 11287: 11233: 11227: 11226: 11225: 11216: 11212: 11203: 11199: 11187: 11181: 11180: 11179: 11170: 11166: 11157: 11153: 11141: 11135: 11134: 11133: 11124: 11120: 11111: 11107: 11095: 11094: 11085: 11084: 11082: 11079: 11078: 11072: 11065: 11058: 11047: 11040: 11033: 10997: 10991: 10990: 10989: 10983: 10979: 10970: 10964: 10963: 10962: 10956: 10952: 10943: 10937: 10936: 10935: 10929: 10925: 10916: 10915: 10913: 10910: 10909: 10906:position vector 10903: 10896: 10889: 10877: 10837: 10831: 10807: 10799: 10797:Vector quantity 10793: 10762: 10757: 10756: 10750: 10745: 10744: 10735: 10731: 10725: 10721: 10712: 10708: 10702: 10698: 10683: 10677: 10676: 10675: 10669: 10665: 10656: 10650: 10649: 10648: 10642: 10638: 10623: 10617: 10616: 10615: 10609: 10605: 10596: 10590: 10589: 10588: 10582: 10578: 10573: 10570: 10569: 10542: 10541: 10532: 10526: 10525: 10524: 10515: 10511: 10502: 10498: 10486: 10480: 10479: 10478: 10469: 10465: 10456: 10452: 10442: 10436: 10430: 10429: 10428: 10419: 10415: 10406: 10402: 10390: 10384: 10383: 10382: 10373: 10369: 10360: 10356: 10350: 10349: 10337: 10331: 10330: 10329: 10323: 10319: 10310: 10304: 10303: 10302: 10296: 10292: 10283: 10277: 10276: 10275: 10269: 10265: 10256: 10250: 10249: 10248: 10242: 10238: 10228: 10219: 10213: 10212: 10211: 10205: 10201: 10192: 10186: 10185: 10184: 10178: 10174: 10165: 10159: 10158: 10157: 10151: 10147: 10138: 10132: 10131: 10130: 10124: 10120: 10113: 10111: 10108: 10107: 10087: 10081: 10080: 10079: 10070: 10066: 10057: 10053: 10041: 10035: 10034: 10033: 10024: 10020: 10011: 10007: 9992: 9986: 9985: 9984: 9978: 9974: 9965: 9959: 9958: 9957: 9951: 9947: 9932: 9926: 9925: 9924: 9918: 9914: 9905: 9899: 9898: 9897: 9891: 9887: 9882: 9879: 9878: 9875: 9799:rotation matrix 9784: 9771: 9764: 9757: 9746: 9739: 9732: 9711: 9710: 9704: 9699: 9698: 9689: 9684: 9683: 9676: 9670: 9666: 9663: 9662: 9656: 9651: 9650: 9641: 9636: 9635: 9628: 9622: 9618: 9615: 9614: 9608: 9603: 9602: 9593: 9588: 9587: 9580: 9574: 9570: 9567: 9566: 9560: 9555: 9554: 9545: 9540: 9539: 9532: 9526: 9522: 9519: 9518: 9512: 9507: 9506: 9497: 9492: 9491: 9484: 9478: 9474: 9471: 9470: 9464: 9459: 9458: 9449: 9444: 9443: 9436: 9430: 9426: 9423: 9422: 9416: 9411: 9410: 9401: 9396: 9395: 9388: 9382: 9378: 9375: 9374: 9368: 9363: 9362: 9353: 9348: 9347: 9340: 9334: 9330: 9327: 9326: 9320: 9315: 9314: 9305: 9300: 9299: 9292: 9286: 9282: 9278: 9276: 9273: 9272: 9256: 9247: 9234: 9168: 9167: 9161: 9160: 9154: 9153: 9143: 9142: 9135: 9134: 9128: 9124: 9122: 9116: 9112: 9110: 9104: 9100: 9097: 9096: 9090: 9086: 9084: 9078: 9074: 9072: 9066: 9062: 9059: 9058: 9052: 9048: 9046: 9040: 9036: 9034: 9028: 9024: 9017: 9016: 9006: 9005: 8999: 8998: 8992: 8991: 8981: 8980: 8978: 8975: 8974: 8956: 8955: 8943: 8939: 8927: 8923: 8911: 8907: 8900: 8894: 8893: 8881: 8877: 8865: 8861: 8849: 8845: 8838: 8832: 8831: 8819: 8815: 8803: 8799: 8787: 8783: 8776: 8769: 8767: 8764: 8763: 8745: 8744: 8735: 8730: 8729: 8720: 8715: 8714: 8702: 8697: 8696: 8687: 8682: 8681: 8669: 8664: 8663: 8654: 8649: 8648: 8638: 8632: 8631: 8622: 8617: 8616: 8607: 8602: 8601: 8589: 8584: 8583: 8574: 8569: 8568: 8556: 8551: 8550: 8541: 8536: 8535: 8525: 8519: 8518: 8509: 8504: 8503: 8494: 8489: 8488: 8476: 8471: 8470: 8461: 8456: 8455: 8443: 8438: 8437: 8428: 8423: 8422: 8412: 8405: 8403: 8400: 8399: 8381: 8380: 8371: 8366: 8365: 8353: 8348: 8347: 8335: 8330: 8329: 8317: 8312: 8311: 8298: 8292: 8291: 8282: 8277: 8276: 8264: 8259: 8258: 8246: 8241: 8240: 8228: 8223: 8222: 8209: 8203: 8202: 8193: 8188: 8187: 8175: 8170: 8169: 8157: 8152: 8151: 8139: 8134: 8133: 8120: 8113: 8111: 8108: 8107: 8032: 8031: 8022: 8017: 8016: 8008: 8001: 7995: 7994: 7985: 7980: 7979: 7971: 7964: 7958: 7957: 7948: 7943: 7942: 7934: 7927: 7920: 7918: 7915: 7914: 7889: 7884: 7883: 7871: 7866: 7865: 7853: 7848: 7847: 7836: 7834: 7831: 7830: 7818: 7811: 7804: 7779: 7778: 7769: 7764: 7763: 7755: 7748: 7742: 7741: 7732: 7727: 7726: 7718: 7711: 7705: 7704: 7695: 7690: 7689: 7681: 7674: 7667: 7665: 7662: 7661: 7633: 7628: 7627: 7615: 7610: 7609: 7597: 7592: 7591: 7580: 7578: 7575: 7574: 7562: 7555: 7548: 7536: 7510: 7502: 7494: 7477: 7469: 7461: 7444: 7436: 7428: 7411: 7403: 7395: 7381: 7373: 7365: 7351: 7343: 7335: 7330: 7327: 7326: 7306: 7305: 7299: 7295: 7293: 7287: 7283: 7281: 7275: 7271: 7268: 7267: 7261: 7257: 7255: 7249: 7245: 7243: 7237: 7233: 7230: 7229: 7223: 7219: 7217: 7211: 7207: 7205: 7199: 7195: 7188: 7187: 7176: 7168: 7160: 7155: 7152: 7151: 7138:In components ( 7092: 7084: 7073: 7062: 7054: 7046: 7041: 7038: 7037: 7010: 7004: 6973: 6967: 6966: 6965: 6956: 6952: 6946: 6942: 6933: 6929: 6923: 6919: 6907: 6901: 6900: 6899: 6890: 6886: 6880: 6876: 6867: 6863: 6857: 6853: 6841: 6835: 6834: 6833: 6824: 6820: 6814: 6810: 6801: 6797: 6791: 6787: 6775: 6774: 6765: 6764: 6762: 6759: 6758: 6733:right-hand rule 6731:). This is the 6615: 6590: 6586: 6577: 6573: 6565: 6557: 6555: 6552: 6551: 6514: 6508: 6484: 6480: 6474: 6470: 6461: 6457: 6451: 6447: 6438: 6434: 6428: 6424: 6416: 6408: 6406: 6403: 6402: 6323: 6319: 6310: 6306: 6298: 6290: 6288: 6285: 6284: 6255:of two vectors 6249: 6243: 6224: 6190: 6189: 6187: 6184: 6183: 6179: 6172: 6166: 6145: 6140: 6139: 6128: 6124: 6118: 6114: 6112: 6103: 6098: 6097: 6086: 6082: 6076: 6072: 6070: 6061: 6056: 6055: 6044: 6040: 6034: 6030: 6028: 6014: 6010: 6005: 6003: 5989: 5988: 5986: 5983: 5982: 5972: 5965: 5958: 5948: 5930: 5909: 5884: 5876: 5874: 5862: 5858: 5856: 5853: 5852: 5841: 5834: 5827: 5794: 5789: 5776: 5771: 5758: 5753: 5747: 5735: 5731: 5729: 5726: 5725: 5697:is denoted by ‖ 5673: 5535: 5530: 5529: 5520: 5516: 5501: 5496: 5495: 5486: 5482: 5467: 5462: 5461: 5452: 5448: 5434: 5429: 5426: 5425: 5385: 5379: 5322:to the head of 5281: 5276: 5275: 5266: 5262: 5253: 5249: 5237: 5232: 5231: 5222: 5218: 5209: 5205: 5193: 5188: 5187: 5178: 5174: 5165: 5161: 5150: 5142: 5140: 5137: 5136: 5012:to the head of 4966: 4961: 4960: 4951: 4947: 4938: 4934: 4922: 4917: 4916: 4907: 4903: 4894: 4890: 4878: 4873: 4872: 4863: 4859: 4850: 4846: 4835: 4827: 4825: 4822: 4821: 4810: 4804: 4787:equidirectional 4762: 4758: 4746: 4742: 4732: 4728: 4716: 4712: 4702: 4698: 4686: 4682: 4680: 4677: 4676: 4655: 4649: 4648: 4647: 4641: 4637: 4628: 4622: 4621: 4620: 4614: 4610: 4601: 4595: 4594: 4593: 4587: 4583: 4574: 4573: 4571: 4568: 4567: 4546: 4540: 4539: 4538: 4532: 4528: 4519: 4513: 4512: 4511: 4505: 4501: 4492: 4486: 4485: 4484: 4478: 4474: 4465: 4464: 4462: 4459: 4458: 4445: 4420: 4416: 4407: 4403: 4393: 4389: 4380: 4376: 4366: 4362: 4353: 4349: 4347: 4344: 4343: 4326: 4320: 4319: 4318: 4312: 4308: 4299: 4293: 4292: 4291: 4285: 4281: 4272: 4266: 4265: 4264: 4258: 4254: 4245: 4244: 4242: 4239: 4238: 4221: 4215: 4214: 4213: 4207: 4203: 4194: 4188: 4187: 4186: 4180: 4176: 4167: 4161: 4160: 4159: 4153: 4149: 4140: 4139: 4137: 4134: 4133: 4130: 4106: 4100: 4099: 4098: 4092: 4088: 4079: 4073: 4072: 4071: 4065: 4061: 4052: 4046: 4045: 4044: 4038: 4034: 4025: 4024: 4022: 4019: 4018: 3973: 3967: 3966: 3965: 3929: 3923: 3922: 3921: 3885: 3879: 3878: 3877: 3875: 3872: 3871: 3864: 3858: 3774: 3773: 3759: 3758: 3744: 3743: 3741: 3738: 3737: 3711: 3710: 3696: 3695: 3681: 3680: 3678: 3675: 3674: 3644: 3643: 3629: 3628: 3614: 3613: 3611: 3608: 3607: 3570: 3564: 3549: 3520: 3519: 3513: 3509: 3500: 3499: 3493: 3489: 3480: 3479: 3473: 3469: 3460: 3455: 3454: 3445: 3440: 3439: 3430: 3425: 3424: 3416: 3414: 3411: 3410: 3406: 3397: 3388: 3378: 3371: 3364: 3336: 3335: 3333: 3330: 3329: 3325:, in which the 3304: 3303: 3289: 3288: 3274: 3273: 3271: 3268: 3267: 3251: 3243: 3235: 3233: 3230: 3229: 3220: 3213: 3206: 3174:are called the 3173: 3166: 3159: 3132: 3126: 3125: 3124: 3118: 3114: 3105: 3099: 3098: 3097: 3091: 3087: 3078: 3072: 3071: 3070: 3064: 3060: 3051: 3046: 3045: 3036: 3031: 3030: 3021: 3016: 3015: 3007: 3005: 3002: 3001: 2955: 2951: 2921: 2917: 2887: 2883: 2871: 2867: 2858: 2854: 2845: 2841: 2830: 2828: 2825: 2824: 2818: 2754: 2748: 2747: 2746: 2710: 2704: 2703: 2702: 2666: 2660: 2659: 2658: 2656: 2653: 2652: 2622: 2618: 2612: 2608: 2599: 2595: 2586: 2582: 2569: 2568: 2562: 2558: 2555: 2554: 2548: 2544: 2541: 2540: 2534: 2530: 2523: 2522: 2514: 2512: 2509: 2508: 2471: 2467: 2452: 2448: 2433: 2429: 2420: 2416: 2407: 2403: 2392: 2390: 2387: 2386: 2380: 2351: 2347: 2338: 2334: 2325: 2321: 2310: 2308: 2305: 2304: 2281: 2277: 2268: 2264: 2255: 2251: 2240: 2238: 2235: 2234: 2228: 2197: 2195: 2193: 2190: 2189: 2150: 2148: 2145: 2144: 1974: 1973: 1971: 1968: 1967: 1934: 1926: 1924: 1923: 1921: 1918: 1917: 1882: 1879: 1874: 1872: 1869: 1868: 1836: 1835: 1833: 1830: 1829: 1799: 1797: 1794: 1793: 1776: 1774: 1771: 1770: 1753: 1751: 1748: 1747: 1725: 1719: 1717:Representations 1676:change of basis 1652: 1650:Generalizations 1625:Minkowski space 1569: 1431: 1428: 1427: 1412: 1398: 1387: 1363: 1361: 1359: 1356: 1355: 1348: 1341: 1334: 1310:linear momentum 1278: 1238: 1223: 1202: 1197: 1196: 1194: 1191: 1190: 1163: 1161: 1159: 1156: 1155: 1151: 1131: 1111: 1106: 1105: 1103: 1100: 1099: 1091: 1087: 1063: 1058: 1057: 1055: 1052: 1051: 1031: 1026: 1025: 1023: 1020: 1019: 998: 993: 992: 990: 987: 986: 954: 952: 949: 948: 912: 910: 907: 906: 874: 872: 869: 868: 860: 833: 829: 817: 785: 783: 781: 778: 777: 765: 739: 727: 715: 692: 644: 635: 627: 626: 624: 623: 621: 618: 617: 597: 589: 587: 586: 584: 581: 580: 570:resultant force 521:Euclidean space 505:Euclidean space 481: 472:Vector Analysis 411:Matthew O'Brien 395:Augustin Cauchy 375: 364:complex numbers 345: 328: 297: 156: 148: 146: 145: 143: 140: 139: 114:and possibly a 103:vector quantity 39: 28: 23: 22: 15: 12: 11: 5: 13400: 13390: 13389: 13384: 13379: 13374: 13369: 13367:Linear algebra 13364: 13359: 13354: 13337: 13336: 13334: 13333: 13322: 13319: 13318: 13316: 13315: 13310: 13305: 13300: 13295: 13293:Floating-point 13289: 13287: 13281: 13280: 13278: 13277: 13275:Tensor product 13272: 13267: 13262: 13260:Function space 13257: 13252: 13246: 13244: 13237: 13236: 13234: 13233: 13228: 13223: 13218: 13213: 13208: 13203: 13198: 13196:Triple product 13193: 13188: 13182: 13180: 13174: 13173: 13171: 13170: 13165: 13160: 13155: 13150: 13145: 13140: 13134: 13132: 13126: 13125: 13123: 13122: 13117: 13112: 13110:Transformation 13107: 13102: 13100:Multiplication 13097: 13092: 13087: 13082: 13076: 13074: 13068: 13067: 13060: 13058: 13056: 13055: 13050: 13045: 13040: 13035: 13030: 13025: 13020: 13015: 13010: 13005: 13000: 12995: 12990: 12985: 12980: 12975: 12970: 12965: 12959: 12957: 12956:Basic concepts 12953: 12952: 12950: 12949: 12944: 12938: 12935: 12934: 12931:Linear algebra 12927: 12926: 12919: 12912: 12904: 12898: 12897: 12887: 12877: 12857: 12856:External links 12854: 12853: 12852: 12846: 12828: 12822: 12805: 12802: 12801: 12800: 12794: 12774: 12768: 12752: 12742: 12723: 12717: 12700: 12694: 12679: 12673: 12653: 12647: 12627: 12624: 12622: 12619: 12616: 12615: 12600: 12582: 12563: 12556: 12531: 12506: 12460: 12449: 12433:"1.1: Vectors" 12421: 12392: 12384:located vector 12375: 12373:, p. 1678 12363: 12350: 12303: 12278: 12247: 12220: 12213: 12196: 12131: 12116: 12105: 12093: 12092: 12089: 12088: 12078: 12077: 12075: 12072: 12070: 12069: 12064: 12059: 12054: 12049: 12044: 12039: 12033: 12028: 12023: 12018: 12013: 12008: 12003: 11998: 11989: 11987:Function space 11984: 11975: 11970: 11965: 11963:Complex number 11960: 11955: 11950: 11940: 11938: 11935: 11920:magnetic field 11917: 11866: 11854:tangent vector 11819:electric field 11763: 11760: 11756: 11695: 11694: 11677: 11676: 11638: 11636: 11629: 11622: 11621: 11572: 11570: 11563: 11558: 11532: 11531: 11529: 11522: 11516: 11513: 11501: 11498: 11493: 11487: 11481: 11476: 11470: 11464: 11461: 11456: 11451: 11448: 11417: 11412: 11409: 11404: 11383: 11380: 11354: 11339: 11334: 11328: 11322: 11317: 11312: 11309: 11304: 11298: 11241: 11236: 11230: 11224: 11219: 11215: 11211: 11206: 11202: 11198: 11195: 11190: 11184: 11178: 11173: 11169: 11165: 11160: 11156: 11152: 11149: 11144: 11138: 11132: 11127: 11123: 11119: 11114: 11110: 11106: 11103: 11098: 11093: 11088: 11070: 11063: 11056: 11045: 11038: 11031: 11005: 11000: 10994: 10986: 10982: 10978: 10973: 10967: 10959: 10955: 10951: 10946: 10940: 10932: 10928: 10924: 10919: 10901: 10894: 10887: 10876: 10873: 10861:differentiated 10833:Main article: 10830: 10827: 10806: 10803: 10795:Main article: 10792: 10789: 10770: 10765: 10760: 10753: 10748: 10743: 10738: 10734: 10728: 10724: 10720: 10715: 10711: 10705: 10701: 10697: 10694: 10691: 10686: 10680: 10672: 10668: 10664: 10659: 10653: 10645: 10641: 10637: 10634: 10631: 10626: 10620: 10612: 10608: 10604: 10599: 10593: 10585: 10581: 10577: 10540: 10535: 10529: 10523: 10518: 10514: 10510: 10505: 10501: 10497: 10494: 10489: 10483: 10477: 10472: 10468: 10464: 10459: 10455: 10451: 10448: 10445: 10443: 10439: 10433: 10427: 10422: 10418: 10414: 10409: 10405: 10401: 10398: 10393: 10387: 10381: 10376: 10372: 10368: 10363: 10359: 10355: 10352: 10351: 10348: 10345: 10340: 10334: 10326: 10322: 10318: 10313: 10307: 10299: 10295: 10291: 10286: 10280: 10272: 10268: 10264: 10259: 10253: 10245: 10241: 10237: 10234: 10231: 10229: 10227: 10222: 10216: 10208: 10204: 10200: 10195: 10189: 10181: 10177: 10173: 10168: 10162: 10154: 10150: 10146: 10141: 10135: 10127: 10123: 10119: 10116: 10115: 10095: 10090: 10084: 10078: 10073: 10069: 10065: 10060: 10056: 10052: 10049: 10044: 10038: 10032: 10027: 10023: 10019: 10014: 10010: 10006: 10003: 10000: 9995: 9989: 9981: 9977: 9973: 9968: 9962: 9954: 9950: 9946: 9943: 9940: 9935: 9929: 9921: 9917: 9913: 9908: 9902: 9894: 9890: 9886: 9874: 9871: 9856: 9855: 9852: 9849: 9780: 9769: 9762: 9755: 9744: 9737: 9730: 9707: 9702: 9697: 9692: 9687: 9682: 9679: 9677: 9673: 9669: 9665: 9664: 9659: 9654: 9649: 9644: 9639: 9634: 9631: 9629: 9625: 9621: 9617: 9616: 9611: 9606: 9601: 9596: 9591: 9586: 9583: 9581: 9577: 9573: 9569: 9568: 9563: 9558: 9553: 9548: 9543: 9538: 9535: 9533: 9529: 9525: 9521: 9520: 9515: 9510: 9505: 9500: 9495: 9490: 9487: 9485: 9481: 9477: 9473: 9472: 9467: 9462: 9457: 9452: 9447: 9442: 9439: 9437: 9433: 9429: 9425: 9424: 9419: 9414: 9409: 9404: 9399: 9394: 9391: 9389: 9385: 9381: 9377: 9376: 9371: 9366: 9361: 9356: 9351: 9346: 9343: 9341: 9337: 9333: 9329: 9328: 9323: 9318: 9313: 9308: 9303: 9298: 9295: 9293: 9289: 9285: 9281: 9280: 9261:refers to the 9252: 9243: 9230: 9177: 9172: 9166: 9163: 9162: 9159: 9156: 9155: 9152: 9149: 9148: 9146: 9139: 9131: 9127: 9123: 9119: 9115: 9111: 9107: 9103: 9099: 9098: 9093: 9089: 9085: 9081: 9077: 9073: 9069: 9065: 9061: 9060: 9055: 9051: 9047: 9043: 9039: 9035: 9031: 9027: 9023: 9022: 9020: 9015: 9010: 9004: 9001: 9000: 8997: 8994: 8993: 8990: 8987: 8986: 8984: 8954: 8951: 8946: 8942: 8938: 8935: 8930: 8926: 8922: 8919: 8914: 8910: 8906: 8903: 8901: 8899: 8896: 8895: 8892: 8889: 8884: 8880: 8876: 8873: 8868: 8864: 8860: 8857: 8852: 8848: 8844: 8841: 8839: 8837: 8834: 8833: 8830: 8827: 8822: 8818: 8814: 8811: 8806: 8802: 8798: 8795: 8790: 8786: 8782: 8779: 8777: 8775: 8772: 8771: 8743: 8738: 8733: 8728: 8723: 8718: 8713: 8710: 8705: 8700: 8695: 8690: 8685: 8680: 8677: 8672: 8667: 8662: 8657: 8652: 8647: 8644: 8641: 8639: 8637: 8634: 8633: 8630: 8625: 8620: 8615: 8610: 8605: 8600: 8597: 8592: 8587: 8582: 8577: 8572: 8567: 8564: 8559: 8554: 8549: 8544: 8539: 8534: 8531: 8528: 8526: 8524: 8521: 8520: 8517: 8512: 8507: 8502: 8497: 8492: 8487: 8484: 8479: 8474: 8469: 8464: 8459: 8454: 8451: 8446: 8441: 8436: 8431: 8426: 8421: 8418: 8415: 8413: 8411: 8408: 8407: 8379: 8374: 8369: 8364: 8361: 8356: 8351: 8346: 8343: 8338: 8333: 8328: 8325: 8320: 8315: 8310: 8307: 8304: 8301: 8299: 8297: 8294: 8293: 8290: 8285: 8280: 8275: 8272: 8267: 8262: 8257: 8254: 8249: 8244: 8239: 8236: 8231: 8226: 8221: 8218: 8215: 8212: 8210: 8208: 8205: 8204: 8201: 8196: 8191: 8186: 8183: 8178: 8173: 8168: 8165: 8160: 8155: 8150: 8147: 8142: 8137: 8132: 8129: 8126: 8123: 8121: 8119: 8116: 8115: 8047:The values of 8030: 8025: 8020: 8015: 8011: 8007: 8004: 8002: 8000: 7997: 7996: 7993: 7988: 7983: 7978: 7974: 7970: 7967: 7965: 7963: 7960: 7959: 7956: 7951: 7946: 7941: 7937: 7933: 7930: 7928: 7926: 7923: 7922: 7892: 7887: 7882: 7879: 7874: 7869: 7864: 7861: 7856: 7851: 7846: 7843: 7839: 7816: 7809: 7802: 7777: 7772: 7767: 7762: 7758: 7754: 7751: 7749: 7747: 7744: 7743: 7740: 7735: 7730: 7725: 7721: 7717: 7714: 7712: 7710: 7707: 7706: 7703: 7698: 7693: 7688: 7684: 7680: 7677: 7675: 7673: 7670: 7669: 7641: 7636: 7631: 7626: 7623: 7618: 7613: 7608: 7605: 7600: 7595: 7590: 7587: 7583: 7560: 7553: 7546: 7535: 7532: 7520: 7517: 7513: 7505: 7497: 7493: 7490: 7487: 7484: 7480: 7472: 7464: 7460: 7457: 7454: 7451: 7447: 7439: 7431: 7427: 7424: 7421: 7418: 7414: 7406: 7398: 7394: 7391: 7388: 7384: 7376: 7368: 7364: 7361: 7358: 7354: 7346: 7338: 7334: 7310: 7302: 7298: 7294: 7290: 7286: 7282: 7278: 7274: 7270: 7269: 7264: 7260: 7256: 7252: 7248: 7244: 7240: 7236: 7232: 7231: 7226: 7222: 7218: 7214: 7210: 7206: 7202: 7198: 7194: 7193: 7191: 7186: 7183: 7179: 7171: 7163: 7159: 7146:of the 3-by-3 7117:parallelepiped 7102: 7099: 7095: 7091: 7087: 7083: 7080: 7076: 7072: 7069: 7065: 7057: 7049: 7045: 7006:Main article: 7003: 7000: 6981: 6976: 6970: 6964: 6959: 6955: 6949: 6945: 6941: 6936: 6932: 6926: 6922: 6918: 6915: 6910: 6904: 6898: 6893: 6889: 6883: 6879: 6875: 6870: 6866: 6860: 6856: 6852: 6849: 6844: 6838: 6832: 6827: 6823: 6817: 6813: 6809: 6804: 6800: 6794: 6790: 6786: 6783: 6778: 6773: 6768: 6738:The length of 6618: 6613: 6610: 6607: 6604: 6601: 6597: 6593: 6589: 6584: 6580: 6576: 6572: 6568: 6564: 6560: 6522:vector product 6510:Main article: 6507: 6504: 6492: 6487: 6483: 6477: 6473: 6469: 6464: 6460: 6454: 6450: 6446: 6441: 6437: 6431: 6427: 6423: 6419: 6415: 6411: 6343: 6340: 6337: 6334: 6330: 6326: 6322: 6317: 6313: 6309: 6305: 6301: 6297: 6293: 6271:scalar product 6245:Main article: 6242: 6239: 6197: 6194: 6168:Main article: 6165: 6162: 6148: 6143: 6135: 6131: 6127: 6121: 6117: 6111: 6106: 6101: 6093: 6089: 6085: 6079: 6075: 6069: 6064: 6059: 6051: 6047: 6043: 6037: 6033: 6027: 6021: 6017: 6013: 6008: 6002: 5996: 5993: 5970: 5963: 5956: 5926:Main article: 5908: 5905: 5893: 5887: 5883: 5879: 5873: 5869: 5865: 5861: 5839: 5832: 5825: 5804: 5797: 5792: 5788: 5784: 5779: 5774: 5770: 5766: 5761: 5756: 5752: 5746: 5742: 5738: 5734: 5719:Euclidean norm 5707:absolute value 5693:of the vector 5672: 5669: 5543: 5538: 5533: 5528: 5523: 5519: 5515: 5512: 5509: 5504: 5499: 5494: 5489: 5485: 5481: 5478: 5475: 5470: 5465: 5460: 5455: 5451: 5447: 5444: 5441: 5437: 5433: 5381:Main article: 5378: 5375: 5289: 5284: 5279: 5274: 5269: 5265: 5261: 5256: 5252: 5248: 5245: 5240: 5235: 5230: 5225: 5221: 5217: 5212: 5208: 5204: 5201: 5196: 5191: 5186: 5181: 5177: 5173: 5168: 5164: 5160: 5157: 5153: 5149: 5145: 4974: 4969: 4964: 4959: 4954: 4950: 4946: 4941: 4937: 4933: 4930: 4925: 4920: 4915: 4910: 4906: 4902: 4897: 4893: 4889: 4886: 4881: 4876: 4871: 4866: 4862: 4858: 4853: 4849: 4845: 4842: 4838: 4834: 4830: 4803: 4800: 4770: 4765: 4761: 4757: 4754: 4749: 4745: 4740: 4735: 4731: 4727: 4724: 4719: 4715: 4710: 4705: 4701: 4697: 4694: 4689: 4685: 4658: 4652: 4644: 4640: 4636: 4631: 4625: 4617: 4613: 4609: 4604: 4598: 4590: 4586: 4582: 4577: 4549: 4543: 4535: 4531: 4527: 4522: 4516: 4508: 4504: 4500: 4495: 4489: 4481: 4477: 4473: 4468: 4444: 4441: 4428: 4423: 4419: 4415: 4410: 4406: 4401: 4396: 4392: 4388: 4383: 4379: 4374: 4369: 4365: 4361: 4356: 4352: 4329: 4323: 4315: 4311: 4307: 4302: 4296: 4288: 4284: 4280: 4275: 4269: 4261: 4257: 4253: 4248: 4224: 4218: 4210: 4206: 4202: 4197: 4191: 4183: 4179: 4175: 4170: 4164: 4156: 4152: 4148: 4143: 4129: 4126: 4114: 4109: 4103: 4095: 4091: 4087: 4082: 4076: 4068: 4064: 4060: 4055: 4049: 4041: 4037: 4033: 4028: 4002: 3999: 3996: 3993: 3990: 3987: 3984: 3981: 3976: 3970: 3961: 3958: 3955: 3952: 3949: 3946: 3943: 3940: 3937: 3932: 3926: 3917: 3914: 3911: 3908: 3905: 3902: 3899: 3896: 3893: 3888: 3882: 3857: 3854: 3781: 3778: 3772: 3766: 3763: 3757: 3751: 3748: 3718: 3715: 3709: 3703: 3700: 3694: 3688: 3685: 3651: 3648: 3642: 3636: 3633: 3627: 3621: 3618: 3563: 3560: 3552:index notation 3545: 3528: 3523: 3516: 3512: 3508: 3503: 3496: 3492: 3488: 3483: 3476: 3472: 3468: 3463: 3458: 3453: 3448: 3443: 3438: 3433: 3428: 3423: 3419: 3402: 3393: 3384: 3376: 3369: 3362: 3341: 3311: 3308: 3302: 3296: 3293: 3287: 3281: 3278: 3254: 3250: 3246: 3242: 3238: 3218: 3211: 3204: 3171: 3164: 3157: 3140: 3135: 3129: 3121: 3117: 3113: 3108: 3102: 3094: 3090: 3086: 3081: 3075: 3067: 3063: 3059: 3054: 3049: 3044: 3039: 3034: 3029: 3024: 3019: 3014: 3010: 2984: 2981: 2978: 2975: 2972: 2969: 2966: 2963: 2958: 2954: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2924: 2920: 2916: 2913: 2910: 2907: 2904: 2901: 2898: 2895: 2890: 2886: 2882: 2879: 2874: 2870: 2866: 2861: 2857: 2853: 2848: 2844: 2840: 2837: 2833: 2786: 2783: 2780: 2777: 2774: 2771: 2768: 2765: 2762: 2757: 2751: 2742: 2739: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2713: 2707: 2698: 2695: 2692: 2689: 2686: 2683: 2680: 2677: 2674: 2669: 2663: 2649:standard basis 2630: 2625: 2621: 2615: 2611: 2602: 2598: 2589: 2585: 2581: 2578: 2573: 2565: 2561: 2557: 2556: 2551: 2547: 2543: 2542: 2537: 2533: 2529: 2528: 2526: 2521: 2517: 2507:, as follows: 2482: 2479: 2474: 2470: 2466: 2461: 2458: 2455: 2451: 2447: 2444: 2441: 2436: 2432: 2428: 2423: 2419: 2415: 2410: 2406: 2402: 2399: 2395: 2362: 2359: 2354: 2350: 2346: 2341: 2337: 2333: 2328: 2324: 2320: 2317: 2313: 2303:also written, 2292: 2289: 2284: 2280: 2276: 2271: 2267: 2263: 2258: 2254: 2250: 2247: 2243: 2207: 2203: 2200: 2175: 2172: 2169: 2166: 2163: 2160: 2157: 2153: 2106:real numbers ( 2035:terminal point 2021:is called the 2001:is called the 1977: 1937: 1932: 1929: 1885: 1881: 1878: 1843: 1840: 1802: 1779: 1756: 1718: 1715: 1651: 1648: 1644:thermodynamics 1568: 1565: 1553: 1549: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1522: 1519: 1516: 1513: 1510: 1507: 1504: 1501: 1498: 1495: 1492: 1489: 1486: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1450: 1447: 1444: 1441: 1438: 1435: 1373: 1369: 1366: 1333: 1330: 1322:magnetic field 1277: 1274: 1237: 1234: 1210: 1205: 1200: 1178: 1173: 1169: 1166: 1119: 1114: 1109: 1066: 1061: 1034: 1029: 1001: 996: 961: 958: 924: 919: 916: 903:additive group 886: 881: 878: 857:linear algebra 800: 795: 791: 788: 691: 688: 648: 641: 638: 633: 630: 601: 595: 592: 550:terminal point 480: 477: 382:imaginary part 368:imaginary unit 296: 293: 238:distributivity 222:multiplication 166: 159: 154: 151: 133:terminal point 82:spatial vector 26: 9: 6: 4: 3: 2: 13399: 13388: 13385: 13383: 13380: 13378: 13375: 13373: 13370: 13368: 13365: 13363: 13360: 13358: 13355: 13353: 13350: 13349: 13347: 13332: 13324: 13323: 13320: 13314: 13311: 13309: 13308:Sparse matrix 13306: 13304: 13301: 13299: 13296: 13294: 13291: 13290: 13288: 13286: 13282: 13276: 13273: 13271: 13268: 13266: 13263: 13261: 13258: 13256: 13253: 13251: 13248: 13247: 13245: 13243:constructions 13242: 13238: 13232: 13231:Outermorphism 13229: 13227: 13224: 13222: 13219: 13217: 13214: 13212: 13209: 13207: 13204: 13202: 13199: 13197: 13194: 13192: 13191:Cross product 13189: 13187: 13184: 13183: 13181: 13179: 13175: 13169: 13166: 13164: 13161: 13159: 13158:Outer product 13156: 13154: 13151: 13149: 13146: 13144: 13141: 13139: 13138:Orthogonality 13136: 13135: 13133: 13131: 13127: 13121: 13118: 13116: 13115:Cramer's rule 13113: 13111: 13108: 13106: 13103: 13101: 13098: 13096: 13093: 13091: 13088: 13086: 13085:Decomposition 13083: 13081: 13078: 13077: 13075: 13073: 13069: 13064: 13054: 13051: 13049: 13046: 13044: 13041: 13039: 13036: 13034: 13031: 13029: 13026: 13024: 13021: 13019: 13016: 13014: 13011: 13009: 13006: 13004: 13001: 12999: 12996: 12994: 12991: 12989: 12986: 12984: 12981: 12979: 12976: 12974: 12971: 12969: 12966: 12964: 12961: 12960: 12958: 12954: 12948: 12945: 12943: 12940: 12939: 12936: 12932: 12925: 12920: 12918: 12913: 12911: 12906: 12905: 12902: 12895: 12891: 12888: 12885: 12881: 12878: 12874: 12870: 12869: 12864: 12860: 12859: 12849: 12843: 12839: 12838: 12833: 12829: 12825: 12819: 12815: 12814: 12808: 12807: 12797: 12795:0-486-65812-0 12791: 12786: 12785: 12779: 12778:Pedoe, Daniel 12775: 12771: 12769:0-387-96205-0 12765: 12761: 12757: 12753: 12748: 12743: 12739: 12735: 12734: 12729: 12724: 12720: 12714: 12710: 12706: 12701: 12697: 12695:1-55369-133-4 12691: 12687: 12686: 12680: 12676: 12670: 12665: 12664: 12658: 12654: 12650: 12644: 12640: 12639: 12634: 12630: 12629: 12611: 12607: 12603: 12601:9781563479274 12597: 12593: 12586: 12579: 12574: 12572: 12570: 12568: 12559: 12557:0-387-94746-9 12553: 12549: 12548: 12540: 12538: 12536: 12521:on 2007-01-22 12520: 12516: 12510: 12502: 12496: 12490: 12484: 12478: 12473: 12469: 12464: 12458: 12453: 12438: 12434: 12428: 12426: 12418: 12414: 12408: 12404: 12396: 12389: 12385: 12379: 12372: 12367: 12360: 12354: 12335: 12328: 12323: 12319: 12314: 12312: 12310: 12308: 12293: 12289: 12282: 12268: 12264: 12258: 12256: 12254: 12252: 12237: 12233: 12227: 12225: 12216: 12214:9780195219425 12210: 12206: 12200: 12186: 12183:Jeff Miller. 12178: 12170: 12166: 12165: 12159: 12157: 12149: 12145: 12141: 12135: 12129: 12125: 12120: 12114: 12109: 12103: 12098: 12094: 12083: 12079: 12068: 12065: 12063: 12060: 12058: 12055: 12053: 12052:Vector bundle 12050: 12048: 12045: 12043: 12040: 12038:(of a vector) 12037: 12034: 12032: 12029: 12027: 12024: 12022: 12019: 12017: 12014: 12012: 12009: 12007: 12006:Normal vector 12004: 12002: 12001:Hilbert space 11999: 11997: 11993: 11990: 11988: 11985: 11983: 11979: 11976: 11974: 11971: 11969: 11966: 11964: 11961: 11959: 11956: 11954: 11951: 11949: 11945: 11942: 11941: 11934: 11932: 11927: 11925: 11921: 11915: 11913: 11909: 11905: 11901: 11896: 11894: 11893:cross product 11890: 11889:polar vectors 11886: 11882: 11878: 11877: 11872: 11871: 11864: 11861: 11859: 11855: 11851: 11850:contravariant 11847: 11843: 11839: 11834: 11832: 11828: 11824: 11820: 11816: 11812: 11808: 11803: 11799: 11795: 11791: 11787: 11783: 11778: 11761: 11758: 11755: 11747: 11743: 11738: 11733: 11730: 11726: 11721: 11717: 11714: 11710: 11706: 11702: 11691: 11688: 11673: 11670: 11662: 11659:December 2021 11652: 11648: 11642: 11639:This section 11637: 11628: 11627: 11618: 11615: 11607: 11604:December 2021 11597: 11593: 11589: 11583: 11582: 11578: 11573:This section 11571: 11567: 11562: 11561: 11556: 11554: 11547: 11546: 11541: 11540: 11535: 11530: 11521: 11520: 11512: 11499: 11491: 11479: 11474: 11459: 11449: 11446: 11439: 11435: 11430: 11410: 11407: 11392: 11388: 11379: 11377: 11373: 11370: 11366: 11364: 11360: 11353: 11337: 11332: 11320: 11310: 11307: 11302: 11285: 11281: 11277: 11274: 11269: 11267: 11263: 11259: 11255: 11239: 11234: 11217: 11213: 11209: 11204: 11200: 11193: 11188: 11171: 11167: 11163: 11158: 11154: 11147: 11142: 11125: 11121: 11117: 11112: 11108: 11101: 11091: 11076: 11069: 11062: 11055: 11051: 11044: 11037: 11030: 11026: 11021: 11019: 11003: 10998: 10984: 10980: 10976: 10971: 10957: 10953: 10949: 10944: 10930: 10926: 10922: 10907: 10900: 10893: 10886: 10882: 10872: 10870: 10866: 10862: 10858: 10854: 10850: 10846: 10842: 10836: 10826: 10824: 10819: 10815: 10812: 10811:dimensionless 10802: 10798: 10788: 10786: 10781: 10768: 10763: 10751: 10736: 10732: 10726: 10722: 10718: 10713: 10709: 10703: 10699: 10692: 10684: 10670: 10666: 10662: 10657: 10643: 10639: 10632: 10624: 10610: 10606: 10602: 10597: 10583: 10579: 10568: 10564: 10560: 10555: 10538: 10533: 10516: 10512: 10508: 10503: 10499: 10492: 10487: 10470: 10466: 10462: 10457: 10453: 10446: 10444: 10437: 10420: 10416: 10412: 10407: 10403: 10396: 10391: 10374: 10370: 10366: 10361: 10357: 10346: 10338: 10324: 10320: 10316: 10311: 10297: 10293: 10289: 10284: 10270: 10266: 10262: 10257: 10243: 10239: 10232: 10230: 10220: 10206: 10202: 10198: 10193: 10179: 10175: 10171: 10166: 10152: 10148: 10144: 10139: 10125: 10121: 10093: 10088: 10071: 10067: 10063: 10058: 10054: 10047: 10042: 10025: 10021: 10017: 10012: 10008: 10001: 9993: 9979: 9975: 9971: 9966: 9952: 9948: 9941: 9933: 9919: 9915: 9911: 9906: 9892: 9888: 9870: 9867: 9865: 9861: 9853: 9850: 9847: 9846: 9845: 9842: 9840: 9836: 9832: 9828: 9824: 9820: 9816: 9812: 9808: 9804: 9800: 9796: 9792: 9788: 9783: 9779: 9775: 9768: 9761: 9754: 9751:basis and to 9750: 9743: 9736: 9729: 9724: 9705: 9695: 9690: 9680: 9678: 9671: 9667: 9657: 9647: 9642: 9632: 9630: 9623: 9619: 9609: 9599: 9594: 9584: 9582: 9575: 9571: 9561: 9551: 9546: 9536: 9534: 9527: 9523: 9513: 9503: 9498: 9488: 9486: 9479: 9475: 9465: 9455: 9450: 9440: 9438: 9431: 9427: 9417: 9407: 9402: 9392: 9390: 9383: 9379: 9369: 9359: 9354: 9344: 9342: 9335: 9331: 9321: 9311: 9306: 9296: 9294: 9287: 9283: 9270: 9269:. Therefore, 9268: 9264: 9260: 9255: 9251: 9246: 9242: 9238: 9233: 9229: 9225: 9221: 9217: 9213: 9209: 9205: 9201: 9197: 9193: 9188: 9175: 9170: 9164: 9157: 9150: 9144: 9137: 9129: 9125: 9117: 9113: 9105: 9101: 9091: 9087: 9079: 9075: 9067: 9063: 9053: 9049: 9041: 9037: 9029: 9025: 9018: 9013: 9008: 9002: 8995: 8988: 8982: 8972: 8969: 8952: 8949: 8944: 8940: 8936: 8933: 8928: 8924: 8920: 8917: 8912: 8908: 8904: 8902: 8897: 8890: 8887: 8882: 8878: 8874: 8871: 8866: 8862: 8858: 8855: 8850: 8846: 8842: 8840: 8835: 8828: 8825: 8820: 8816: 8812: 8809: 8804: 8800: 8796: 8793: 8788: 8784: 8780: 8778: 8773: 8761: 8758: 8741: 8736: 8726: 8721: 8711: 8708: 8703: 8693: 8688: 8678: 8675: 8670: 8660: 8655: 8645: 8642: 8640: 8635: 8628: 8623: 8613: 8608: 8598: 8595: 8590: 8580: 8575: 8565: 8562: 8557: 8547: 8542: 8532: 8529: 8527: 8522: 8515: 8510: 8500: 8495: 8485: 8482: 8477: 8467: 8462: 8452: 8449: 8444: 8434: 8429: 8419: 8416: 8414: 8409: 8397: 8394: 8377: 8372: 8362: 8354: 8344: 8341: 8336: 8326: 8323: 8318: 8308: 8302: 8300: 8295: 8288: 8283: 8273: 8265: 8255: 8252: 8247: 8237: 8234: 8229: 8219: 8213: 8211: 8206: 8199: 8194: 8184: 8176: 8166: 8163: 8158: 8148: 8145: 8140: 8130: 8124: 8122: 8117: 8105: 8103: 8099: 8095: 8091: 8087: 8083: 8079: 8074: 8070: 8066: 8062: 8058: 8054: 8050: 8045: 8028: 8023: 8013: 8005: 8003: 7998: 7991: 7986: 7976: 7968: 7966: 7961: 7954: 7949: 7939: 7931: 7929: 7924: 7912: 7910: 7905: 7890: 7880: 7877: 7872: 7862: 7859: 7854: 7844: 7841: 7828: 7826: 7823:, the vector 7822: 7815: 7808: 7801: 7797: 7792: 7775: 7770: 7760: 7752: 7750: 7745: 7738: 7733: 7723: 7715: 7713: 7708: 7701: 7696: 7686: 7678: 7676: 7671: 7659: 7657: 7652: 7639: 7634: 7624: 7621: 7616: 7606: 7603: 7598: 7588: 7585: 7572: 7570: 7566: 7559: 7552: 7545: 7541: 7531: 7518: 7488: 7485: 7455: 7452: 7422: 7419: 7389: 7359: 7323: 7308: 7300: 7296: 7288: 7284: 7276: 7272: 7262: 7258: 7250: 7246: 7238: 7234: 7224: 7220: 7212: 7208: 7200: 7196: 7189: 7184: 7149: 7145: 7141: 7136: 7134: 7130: 7126: 7122: 7118: 7113: 7100: 7089: 7078: 7070: 7035: 7033: 7030: 7027: 7023: 7019: 7015: 7009: 6999: 6997: 6992: 6979: 6974: 6957: 6953: 6947: 6943: 6939: 6934: 6930: 6924: 6920: 6913: 6908: 6891: 6887: 6881: 6877: 6873: 6868: 6864: 6858: 6854: 6847: 6842: 6825: 6821: 6815: 6811: 6807: 6802: 6798: 6792: 6788: 6781: 6771: 6755: 6753: 6749: 6745: 6742: × 6741: 6736: 6734: 6730: 6726: 6722: 6718: 6715: × 6714: 6710: 6706: 6702: 6699: × 6698: 6689: 6685: 6683: 6679: 6675: 6671: 6667: 6663: 6659: 6655: 6651: 6650:perpendicular 6647: 6643: 6639: 6635: 6630: 6608: 6602: 6599: 6570: 6562: 6549: 6547: 6543: 6539: 6536: × 6535: 6531: 6527: 6526:outer product 6523: 6519: 6518:cross product 6513: 6512:Cross product 6506:Cross product 6503: 6490: 6485: 6481: 6475: 6471: 6467: 6462: 6458: 6452: 6448: 6444: 6439: 6435: 6429: 6425: 6421: 6413: 6400: 6397: 6395: 6391: 6387: 6383: 6379: 6375: 6371: 6367: 6363: 6359: 6354: 6341: 6338: 6335: 6332: 6303: 6295: 6282: 6280: 6277: ∙ 6276: 6272: 6268: 6267: 6266:inner product 6262: 6258: 6254: 6248: 6238: 6235: 6231: 6227: 6222: 6218: 6214: 6192: 6177: 6171: 6161: 6146: 6119: 6115: 6109: 6104: 6077: 6073: 6067: 6062: 6035: 6031: 6025: 6000: 5980: 5978: 5969: 5962: 5955: 5951: 5945: 5943: 5939: 5935: 5929: 5922: 5918: 5913: 5904: 5891: 5881: 5871: 5850: 5848: 5843: 5838: 5831: 5824: 5820: 5815: 5802: 5795: 5790: 5786: 5782: 5777: 5772: 5768: 5764: 5759: 5754: 5750: 5744: 5723: 5721: 5720: 5715: 5710: 5708: 5704: 5700: 5696: 5692: 5691: 5686: 5685: 5680: 5679: 5668: 5666: 5662: 5658: 5654: 5650: 5646: 5642: 5638: 5635: 5631: 5628: 5624: 5620: 5616: 5612: 5605: 5601: 5597: 5592: 5588: 5586: 5582: 5578: 5573: 5571: 5567: 5563: 5559: 5554: 5541: 5536: 5521: 5517: 5513: 5507: 5502: 5487: 5483: 5479: 5473: 5468: 5453: 5449: 5445: 5439: 5431: 5423: 5421: 5417: 5413: 5409: 5405: 5402: 5398: 5389: 5384: 5372: 5368: 5363: 5359: 5357: 5353: 5349: 5345: 5341: 5337: 5333: 5329: 5325: 5321: 5317: 5313: 5309: 5305: 5300: 5287: 5282: 5267: 5263: 5259: 5254: 5250: 5243: 5238: 5223: 5219: 5215: 5210: 5206: 5199: 5194: 5179: 5175: 5171: 5166: 5162: 5155: 5147: 5134: 5132: 5128: 5123: 5121: 5117: 5113: 5109: 5105: 5101: 5097: 5093: 5089: 5085: 5081: 5077: 5073: 5069: 5065: 5061: 5057: 5056:parallelogram 5053: 5049: 5045: 5038: 5034: 5029: 5025: 5023: 5019: 5015: 5011: 5007: 5003: 4998: 4996: 4992: 4988: 4972: 4967: 4952: 4948: 4944: 4939: 4935: 4928: 4923: 4908: 4904: 4900: 4895: 4891: 4884: 4879: 4864: 4860: 4856: 4851: 4847: 4840: 4832: 4819: 4815: 4809: 4799: 4797: 4793: 4792:codirectional 4789: 4788: 4782: 4768: 4763: 4759: 4755: 4752: 4747: 4743: 4738: 4733: 4729: 4725: 4722: 4717: 4713: 4708: 4703: 4699: 4695: 4692: 4687: 4683: 4674: 4671: 4656: 4642: 4638: 4634: 4629: 4615: 4611: 4607: 4602: 4588: 4584: 4580: 4565: 4562: 4547: 4533: 4529: 4525: 4520: 4506: 4502: 4498: 4493: 4479: 4475: 4471: 4456: 4454: 4450: 4440: 4426: 4421: 4417: 4413: 4408: 4404: 4399: 4394: 4390: 4386: 4381: 4377: 4372: 4367: 4363: 4359: 4354: 4350: 4342:are equal if 4327: 4313: 4309: 4305: 4300: 4286: 4282: 4278: 4273: 4259: 4255: 4251: 4222: 4208: 4204: 4200: 4195: 4181: 4177: 4173: 4168: 4154: 4150: 4146: 4125: 4112: 4107: 4093: 4089: 4085: 4080: 4066: 4062: 4058: 4053: 4039: 4035: 4031: 4016: 3997: 3994: 3991: 3988: 3985: 3979: 3974: 3959: 3953: 3950: 3947: 3944: 3941: 3935: 3930: 3915: 3909: 3906: 3903: 3900: 3897: 3891: 3886: 3869: 3863: 3853: 3851: 3847: 3842: 3840: 3836: 3832: 3831: 3827: 3822: 3821: 3816: 3812: 3808: 3804: 3799: 3796: 3770: 3755: 3735: 3707: 3692: 3672: 3668: 3640: 3625: 3604: 3597: 3593: 3591: 3587: 3583: 3579: 3575: 3572:As explained 3569: 3559: 3557: 3553: 3548: 3544: 3541:The notation 3539: 3526: 3514: 3510: 3506: 3494: 3490: 3486: 3474: 3470: 3466: 3461: 3451: 3446: 3436: 3431: 3421: 3408: 3405: 3401: 3396: 3392: 3387: 3383: 3379: 3372: 3365: 3358: 3328: 3300: 3285: 3248: 3240: 3226: 3224: 3217: 3210: 3203: 3199: 3195: 3191: 3187: 3183: 3179: 3178: 3170: 3163: 3156: 3151: 3138: 3133: 3119: 3115: 3111: 3106: 3092: 3088: 3084: 3079: 3065: 3061: 3057: 3052: 3042: 3037: 3027: 3022: 3012: 2998: 2982: 2976: 2973: 2970: 2967: 2964: 2956: 2952: 2948: 2942: 2939: 2936: 2933: 2930: 2922: 2918: 2914: 2908: 2905: 2902: 2899: 2896: 2888: 2884: 2880: 2872: 2868: 2864: 2859: 2855: 2851: 2846: 2842: 2835: 2821: 2816: 2812: 2808: 2804: 2800: 2784: 2778: 2775: 2772: 2769: 2766: 2760: 2755: 2740: 2734: 2731: 2728: 2725: 2722: 2716: 2711: 2696: 2690: 2687: 2684: 2681: 2678: 2672: 2667: 2650: 2646: 2641: 2628: 2613: 2609: 2600: 2596: 2587: 2583: 2576: 2571: 2563: 2559: 2549: 2545: 2535: 2531: 2524: 2519: 2506: 2502: 2498: 2497:column vector 2493: 2480: 2472: 2468: 2464: 2459: 2456: 2453: 2449: 2445: 2442: 2439: 2434: 2430: 2426: 2421: 2417: 2413: 2408: 2404: 2397: 2383: 2378: 2377:n-dimensional 2373: 2360: 2352: 2348: 2344: 2339: 2335: 2331: 2326: 2322: 2315: 2290: 2282: 2278: 2274: 2269: 2265: 2261: 2256: 2252: 2245: 2231: 2226: 2221: 2205: 2201: 2198: 2186: 2173: 2167: 2164: 2161: 2155: 2142: 2138: 2133: 2131: 2127: 2126: 2121: 2117: 2113: 2109: 2105: 2101: 2097: 2093: 2085: 2078: 2073: 2069: 2067: 2062: 2058: 2057:perpendicular 2050: 2046: 2044: 2040: 2036: 2032: 2028: 2024: 2020: 2016: 2015:initial point 2012: 2008: 2004: 2000: 1996: 1995:line segments 1991: 1965: 1961: 1957: 1956: 1930: 1927: 1915: 1911: 1908:from a point 1907: 1903: 1883: 1880: 1876: 1866: 1862: 1861: 1838: 1827: 1823: 1819: 1818: 1791: 1745: 1738: 1734: 1729: 1724: 1714: 1712: 1708: 1704: 1700: 1695: 1693: 1689: 1685: 1681: 1677: 1673: 1669: 1668:contravariant 1665: 1661: 1660:basis vectors 1657: 1647: 1645: 1641: 1640:affine spaces 1637: 1636:vector spaces 1632: 1630: 1626: 1622: 1617: 1615: 1611: 1610:parallelotope 1608:-dimensional 1607: 1603: 1599: 1598:parallelogram 1595: 1591: 1587: 1586:cross product 1582: 1578: 1574: 1564: 1551: 1544: 1541: 1538: 1535: 1532: 1529: 1523: 1517: 1514: 1511: 1508: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1481: 1475: 1472: 1469: 1466: 1463: 1460: 1454: 1448: 1445: 1442: 1439: 1436: 1423: 1421: 1415: 1409: 1407: 1401: 1396: 1390: 1371: 1367: 1364: 1351: 1344: 1339: 1329: 1327: 1323: 1319: 1315: 1311: 1307: 1303: 1299: 1295: 1291: 1287: 1283: 1273: 1271: 1267: 1263: 1259: 1255: 1251: 1247: 1243: 1233: 1231: 1227: 1208: 1203: 1176: 1171: 1167: 1164: 1150:of any point 1149: 1145: 1141: 1137: 1117: 1112: 1097: 1086:of dimension 1085: 1082: 1064: 1048: 1032: 1017: 999: 985: 979: 977: 959: 956: 946: 942: 938: 922: 917: 914: 904: 900: 884: 879: 876: 866: 858: 853: 851: 847: 843: 839: 825: 821: 816: 811: 798: 793: 789: 786: 773: 769: 763: 759: 758:parallelogram 754: 750: 746: 742: 735: 731: 723: 719: 713: 712:ordered pairs 709: 705: 701: 697: 694:In classical 687: 685: 680: 678: 674: 673:parallelogram 670: 666: 639: 636: 631: 628: 593: 590: 577: 575: 571: 567: 563: 562: 557: 556: 551: 547: 546:initial point 542: 540: 539: 534: 533: 528: 527: 522: 518: 514: 510: 506: 502: 498: 494: 490: 486: 476: 474: 473: 468: 464: 460: 456: 452: 448: 446: 445:cross product 442: 438: 434: 433: 428: 424: 420: 416: 412: 408: 404: 403:August Möbius 400: 396: 390: 385: 383: 378: 373: 369: 365: 361: 357: 353: 350:(also called 348: 344: 339: 335: 331: 326: 323:as part of a 322: 318: 314: 310: 306: 302: 292: 290: 286: 285:pseudovectors 282: 278: 274: 270: 266: 262: 258: 254: 249: 247: 243: 239: 235: 234:associativity 231: 230:commutativity 227: 223: 219: 215: 211: 207: 203: 199: 195: 190: 186: 183:to the point 182: 177: 164: 152: 149: 137: 134: 130: 127: 126:initial point 123: 122: 117: 113: 109: 105: 104: 99: 95: 91: 87: 83: 79: 75: 71: 67: 63: 59: 52: 48: 43: 37: 33: 19: 13241:Vector space 12973:Vector space 12967: 12866: 12836: 12812: 12783: 12759: 12746: 12731: 12704: 12684: 12662: 12657:Apostol, Tom 12637: 12633:Apostol, Tom 12591: 12585: 12546: 12523:. Retrieved 12519:the original 12509: 12500: 12494: 12488: 12482: 12476: 12471: 12463: 12452: 12441:. Retrieved 12439:. 2013-11-07 12436: 12416: 12413:bound vector 12412: 12411:is called a 12406: 12402: 12395: 12390:, p. 9. 12383: 12378: 12366: 12358: 12353: 12341:. Retrieved 12334:the original 12295:. Retrieved 12291: 12281: 12270:. Retrieved 12266: 12239:. Retrieved 12235: 12204: 12199: 12188:. Retrieved 12162: 12155: 12147: 12143: 12134: 12119: 12108: 12097: 12082: 12026:Pseudovector 11995: 11953:Banach space 11944:Affine space 11933:properties. 11928: 11911: 11897: 11888: 11885:true vectors 11884: 11881:axial vector 11880: 11876:pseudovector 11874: 11868: 11862: 11841: 11835: 11831:acceleration 11811:displacement 11801: 11793: 11789: 11785: 11781: 11776: 11745: 11741: 11736: 11731: 11728: 11724: 11719: 11715: 11704: 11698: 11683: 11665: 11656: 11640: 11610: 11601: 11586:Please help 11574: 11550: 11543: 11537: 11536:Please help 11533: 11438:displacement 11431: 11385: 11371: 11369:Acceleration 11367: 11358: 11351: 11283: 11275: 11270: 11265: 11261: 11257: 11256:relative to 11253: 11077:is a vector 11075:displacement 11067: 11060: 11053: 11049: 11042: 11035: 11028: 11024: 11022: 10898: 10891: 10884: 10880: 10878: 10852: 10848: 10844: 10840: 10838: 10808: 10800: 10782: 10567:pseudoscalar 10556: 9876: 9868: 9860:Euler angles 9857: 9843: 9838: 9834: 9830: 9826: 9814: 9810: 9806: 9802: 9794: 9790: 9781: 9777: 9773: 9766: 9759: 9752: 9748: 9741: 9734: 9727: 9725: 9271: 9258: 9253: 9249: 9244: 9240: 9231: 9227: 9223: 9219: 9215: 9211: 9207: 9203: 9199: 9195: 9191: 9189: 8973: 8970: 8762: 8759: 8398: 8395: 8106: 8097: 8093: 8089: 8088:in terms of 8085: 8081: 8077: 8072: 8068: 8064: 8060: 8056: 8052: 8048: 8046: 7913: 7908: 7906: 7829: 7824: 7820: 7813: 7806: 7799: 7795: 7793: 7660: 7655: 7653: 7573: 7568: 7564: 7557: 7550: 7543: 7539: 7537: 7324: 7139: 7137: 7132: 7128: 7124: 7114: 7036: 7031: 7028: 7025: 7021: 7017: 7013: 7011: 6996:pseudovector 6993: 6756: 6751: 6747: 6743: 6739: 6737: 6724: 6720: 6716: 6712: 6708: 6704: 6700: 6696: 6694: 6681: 6677: 6673: 6669: 6665: 6662:right-handed 6657: 6653: 6645: 6641: 6637: 6633: 6631: 6550: 6545: 6541: 6537: 6533: 6525: 6521: 6517: 6515: 6401: 6398: 6393: 6389: 6385: 6381: 6377: 6369: 6365: 6357: 6355: 6283: 6278: 6274: 6270: 6264: 6260: 6256: 6252: 6250: 6233: 6229: 6225: 6220: 6216: 6212: 6175: 6173: 5981: 5979:‖. That is: 5976: 5967: 5960: 5953: 5949: 5946: 5941: 5937: 5933: 5931: 5920: 5916: 5851: 5844: 5836: 5829: 5822: 5816: 5724: 5717: 5713: 5711: 5702: 5698: 5694: 5688: 5682: 5676: 5674: 5664: 5660: 5656: 5652: 5648: 5644: 5640: 5636: 5633: 5629: 5626: 5622: 5618: 5614: 5611:distributive 5608: 5603: 5602:of a vector 5599: 5595: 5584: 5580: 5576: 5574: 5569: 5565: 5561: 5557: 5555: 5424: 5419: 5415: 5411: 5403: 5396: 5394: 5370: 5366: 5355: 5351: 5347: 5343: 5339: 5335: 5331: 5327: 5323: 5319: 5315: 5311: 5307: 5303: 5301: 5135: 5130: 5126: 5124: 5119: 5115: 5111: 5107: 5103: 5099: 5095: 5091: 5087: 5083: 5079: 5075: 5071: 5067: 5063: 5059: 5051: 5047: 5043: 5041: 5036: 5032: 5021: 5017: 5013: 5009: 5005: 5001: 4999: 4994: 4990: 4986: 4817: 4813: 4811: 4808:Vector space 4795: 4791: 4785: 4783: 4675: 4672: 4566: 4563: 4457: 4448: 4446: 4131: 4014: 3865: 3845: 3843: 3824: 3818: 3814: 3810: 3806: 3800: 3797: 3605: 3602: 3589: 3585: 3571: 3546: 3542: 3540: 3409: 3403: 3399: 3394: 3390: 3385: 3381: 3374: 3367: 3360: 3357:unit vectors 3266:instead (or 3227: 3215: 3208: 3201: 3197: 3193: 3189: 3185: 3181: 3175: 3168: 3161: 3154: 3152: 2999: 2819: 2814: 2806: 2802: 2798: 2644: 2642: 2494: 2381: 2376: 2374: 2229: 2224: 2222: 2187: 2140: 2136: 2134: 2129: 2123: 2107: 2103: 2091: 2089: 2076: 2054: 2038: 2034: 2030: 2026: 2022: 2018: 2014: 2010: 2006: 2002: 1998: 1992: 1954: 1953: 1913: 1909: 1906:displacement 1859: 1858: 1816: 1815: 1768: 1741: 1736: 1732: 1703:vector space 1696: 1667: 1663: 1653: 1633: 1618: 1613: 1605: 1576: 1572: 1570: 1424: 1419: 1413: 1410: 1405: 1399: 1394: 1388: 1349: 1342: 1335: 1326:vector field 1302:displacement 1293: 1289: 1279: 1265: 1261: 1257: 1245: 1239: 1080: 1049: 980: 976:translations 945:Affine space 899:group action 854: 850:acceleration 823: 819: 815:line segment 812: 771: 767: 761: 752: 748: 744: 740: 733: 729: 721: 717: 708:equipollence 693: 683: 681: 668: 578: 560: 559: 555:bound vector 554: 553: 549: 545: 543: 537: 536: 531: 530: 525: 524: 517:vector space 482: 470: 462: 458: 449: 430: 429:∇. In 1878, 427:del operator 422: 414: 392: 387: 381: 376: 355: 351: 346: 337: 333: 329: 316: 305:equipollence 298: 273:displacement 261:acceleration 250: 246:vector space 210:real numbers 201: 197: 194:displacement 188: 184: 180: 178: 135: 132: 128: 125: 119: 110:, including 101: 98:vector space 81: 77: 73: 72:or simply a 69: 55: 50: 46: 13221:Multivector 13186:Determinant 13143:Dot product 12988:Linear span 12756:Lang, Serge 12468:Gibbs, J.W. 12417:free vector 12102:Ivanov 2001 12047:Unit vector 12011:Null vector 11978:Four-vector 11870:orientation 9797:", or the " 9267:dot product 9257:. The term 7144:determinant 7018:box product 6253:dot product 6247:Dot product 6241:Dot product 6176:zero vector 6170:Zero vector 6164:Zero vector 5938:normalizing 5934:unit vector 5928:Unit vector 5907:Unit vector 5847:dot product 5401:real number 4812:The sum of 3803:orientation 3582:projections 2809:-axis of a 2116:coordinates 2039:final point 1912:to a point 1699:mathematics 1612:defined by 1594:orientation 1581:dot product 1416:= (0, 0, 0) 1352:= (0, 1, 0) 1345:= (1, 0, 0) 1016:dot product 974:are called 665:equipollent 579:Two arrows 561:free vector 489:engineering 441:dot product 389:quaternion. 343:real number 218:subtraction 66:engineering 58:mathematics 13352:Kinematics 13346:Categories 13255:Direct sum 13090:Invertible 12993:Linear map 12621:References 12525:2007-01-05 12443:2020-08-19 12343:2010-09-04 12297:2020-08-19 12272:2020-08-19 12241:2020-08-19 12190:2007-05-25 12179:required.) 12128:Pedoe 1988 12031:Quaternion 11982:relativity 11912:reflection 11858:chain rule 11539:improve it 10865:integrated 10857:parametric 10855:) gives a 9864:quaternion 6754:as sides. 6729:orthogonal 6676:, namely, 6223:(that is, 3860:See also: 3839:orthogonal 3592:that set. 3586:decomposed 3327:hat symbol 2501:row vector 1705:over some 1096:isomorphic 941:transitive 469:published 380:to be the 325:quaternion 13285:Numerical 13048:Transpose 12873:EMS Press 12816:. Dover. 12788:. Dover. 12738:EMS Press 12709:MIT Press 12610:652389481 12388:Lang 1986 12263:"Vectors" 11992:Grassmann 11805:abstract 11759:− 11575:does not 11545:talk page 11480:− 11460:⋅ 11210:− 11164:− 11118:− 11092:− 10719:− 10633:∧ 9823:transpose 9696:⋅ 9648:⋅ 9600:⋅ 9552:⋅ 9504:⋅ 9456:⋅ 9408:⋅ 9360:⋅ 9312:⋅ 9239:relating 8727:⋅ 8694:⋅ 8661:⋅ 8614:⋅ 8581:⋅ 8548:⋅ 8501:⋅ 8468:⋅ 8435:⋅ 8363:⋅ 8274:⋅ 8185:⋅ 8014:⋅ 7977:⋅ 7940:⋅ 7761:⋅ 7724:⋅ 7687:⋅ 7489:− 7456:− 7423:− 7090:× 7079:⋅ 6940:− 6874:− 6808:− 6772:× 6609:θ 6603:⁡ 6563:× 6414:⋅ 6339:θ 6336:⁡ 6296:⋅ 6196:→ 6180:(0, 0, 0) 5995:^ 5882:⋅ 5684:magnitude 5583:= −1 and 5399:, by any 5260:− 5216:− 5172:− 5148:− 4756:− 4726:− 4696:− 3780:^ 3777:ϕ 3765:^ 3762:θ 3750:^ 3717:^ 3702:^ 3699:ϕ 3687:^ 3684:ρ 3650:^ 3635:^ 3620:^ 3340:^ 3310:^ 3295:^ 3280:^ 2457:− 2443:⋯ 2206:→ 2043:magnitude 1936:⟶ 1884:∼ 1842:→ 1822:Uppercase 1744:lowercase 1664:covariant 1616:vectors. 1530:− 1491:− 1461:− 1372:→ 1172:→ 960:→ 935:which is 918:→ 880:→ 794:→ 682:The term 647:⟶ 600:⟶ 541:vectors. 538:Euclidean 526:geometric 493:magnitude 372:real line 242:Euclidean 158:⟶ 94:direction 86:magnitude 13331:Category 13270:Subspace 13265:Quotient 13216:Bivector 13130:Bilinear 13072:Matrices 12947:Glossary 12863:"Vector" 12780:(1988). 12758:(1986). 12728:"Vector" 12663:Calculus 12659:(1969). 12638:Calculus 12635:(1967). 12470:(1901). 12371:Itô 1993 12288:"Vector" 12154:"vector 12124:Itô 1993 11937:See also 11931:symmetry 11844:to be a 11823:momentum 11815:velocity 11798:velocity 11286:will be 11273:velocity 11073:) their 10869:calculus 10563:bivector 6652:to both 6596:‖ 6588:‖ 6583:‖ 6575:‖ 6364:between 6329:‖ 6321:‖ 6316:‖ 6308:‖ 6134:‖ 6126:‖ 6092:‖ 6084:‖ 6050:‖ 6042:‖ 6020:‖ 6012:‖ 5868:‖ 5860:‖ 5741:‖ 5733:‖ 5046:because 4796:parallel 4449:opposite 4128:Equality 3835:parallel 3830:rotation 3554:and the 2505:matrices 2031:endpoint 1902:distance 1826:matrices 1697:In pure 1672:gradient 1318:electric 1282:velocity 842:velocity 640:′ 632:′ 479:Overview 309:parallel 269:position 257:velocity 226:negation 214:addition 212:such as 12942:Outline 12875:, 2001 11800:, then 11645:Please 11596:removed 11581:sources 10791:Physics 9819:inverse 9772:as the 9747:as the 9235:is the 9214:basis ( 9198:basis ( 9194:in the 7542:basis { 5412:scalars 5334:, with 3841:to it. 3811:tangent 2805:-, and 2059:to the 1964:fraktur 1688:Tensors 1422:-axis. 1408:-axis. 1404:on the 1393:on the 1336:In the 1250:newtons 901:of the 838:scalars 698:(i.e., 532:spatial 485:physics 366:use an 360:classes 295:History 289:tensors 253:physics 204:. Many 131:with a 116:support 62:physics 13226:Tensor 13038:Kernel 12968:Vector 12963:Scalar 12844: 12820: 12792: 12766: 12715: 12692: 12671: 12645: 12608: 12598: 12554: 12386:. See 12211: 12150:, see 12148:vector 12042:Tensor 11948:points 11924:torque 11916:actual 11908:wheels 11879:or an 11846:tensor 11829:, and 11807:vector 11792:, and 11350:where 11018:length 9263:cosine 9222:, and 9206:, and 8059:, and 7508: 7500: 7475: 7467: 7442: 7434: 7409: 7401: 7379: 7371: 7349: 7341: 7174: 7166: 7148:matrix 7060: 7052: 6711:, and 6680:and (− 6644:, and 6632:where 6356:where 5678:length 5671:Length 5663:+ (−1) 5414:(from 5397:scaled 3963: 3919: 3846:global 3826:radius 3815:radial 3809:, and 3807:normal 3578:add up 3380:, and 3196:, and 3153:where 2986: 2744: 2700: 2606: 2593: 2003:origin 1960:German 1692:tensor 1573:length 1312:, and 1270:meters 1248:of 15 1226:-tuple 1136:origin 1134:as an 897:and a 846:forces 762:vector 706:under 684:vector 677:origin 669:ABB′A′ 574:couple 513:vector 495:and a 409:, and 356:vector 352:scalar 317:vector 265:forces 255:: the 236:, and 224:, and 189:vector 92:) and 90:length 74:vector 64:, and 13095:Minor 13080:Block 13018:Basis 12337:(PDF) 12330:(PDF) 12173: 12074:Notes 11827:force 11709:basis 11434:force 11387:Force 11280:speed 10814:scale 9862:or a 9801:from 9789:from 6530:seven 6372:(see 6362:angle 5598:and 2 5416:scale 5306:from 5098:and ( 3732:) or 3667:basis 3665:as a 3574:above 3184:) of 2112:tuple 2098:in a 2066:arrow 2061:plane 2013:, or 1958:. In 1865:tilde 1707:field 1656:tuple 1619:In a 1577:angle 1298:force 1286:speed 1268:of 4 1242:force 1138:. By 943:(See 710:, of 671:is a 566:force 535:, or 507:. In 341:of a 196:from 13250:Dual 13105:Rank 12842:ISBN 12818:ISBN 12790:ISBN 12764:ISBN 12713:ISBN 12690:ISBN 12669:ISBN 12643:ISBN 12606:OCLC 12596:ISBN 12552:ISBN 12498:and 12486:and 12209:ISBN 12181:and 12144:veho 11744:′ = 11727:′ = 11703:. A 11579:any 11577:cite 11436:and 11271:The 10863:and 7131:and 7012:The 6750:and 6723:and 6672:and 6656:and 6640:and 6544:and 6516:The 6380:and 6368:and 6259:and 6251:The 6174:The 5690:norm 5675:The 5643:and 5625:) = 5369:and 5344:(-b) 5336:(-b) 5328:(-b) 5314:and 5129:and 5106:) + 5070:and 5058:and 5050:and 5035:and 4993:and 4816:and 4790:(or 4564:and 4237:and 3817:and 3180:(or 2128:(or 2023:head 2011:base 2007:tail 1792:and 1592:and 1590:area 1347:and 1320:and 1254:axis 939:and 937:free 848:and 726:and 616:and 572:and 548:and 511:, a 487:and 443:and 287:and 259:and 100:. A 88:(or 68:, a 12884:PDF 11994:'s 11904:car 11887:or 11865:and 11848:of 11649:to 11590:by 11264:to 11052:= ( 11048:), 11027:= ( 10883:= ( 9841:". 9837:to 9829:to 9813:to 9805:to 9793:to 9248:to 7798:= { 7020:or 6684:). 6666:two 6600:sin 6524:or 6333:cos 6237:). 6219:is 5952:= ( 5687:or 5681:or 5575:If 5133:is 5122:). 5114:+ ( 4989:of 3852:). 3828:of 3588:or 3000:or 2817:in 2801:-, 2499:or 2385:). 2223:In 2037:or 2027:tip 1952:or 1904:or 1857:or 1820:. ( 1735:to 1686:). 1666:or 1631:). 1402:= 1 1391:= 1 1094:is 1081:the 905:of 832:to 576:). 483:In 457:'s 271:or 208:on 200:to 80:or 56:In 49:to 13348:: 12871:, 12865:, 12736:, 12730:, 12711:, 12604:. 12566:^ 12534:^ 12435:. 12424:^ 12405:, 12320:, 12306:^ 12290:. 12265:. 12250:^ 12234:. 12223:^ 12161:. 12156:n. 11922:, 11860:. 11833:. 11825:, 11821:, 11817:, 11813:, 11788:, 11548:. 11066:, 11059:, 11041:, 11034:, 11020:. 10897:, 10890:, 10783:A 9782:jk 9765:, 9758:, 9740:, 9733:, 9672:33 9624:32 9576:31 9528:23 9480:22 9432:21 9384:13 9336:12 9288:11 9232:jk 9218:, 9130:33 9118:32 9106:31 9092:23 9080:22 9068:21 9054:13 9042:12 9030:11 8945:33 8929:32 8913:31 8883:23 8867:22 8851:21 8821:13 8805:12 8789:11 8096:, 8092:, 8084:, 8080:, 8067:, 8063:, 8055:, 8051:, 7812:, 7805:, 7556:, 7549:, 7127:, 6735:. 6707:, 6396:. 6279:b, 6232:= 6228:+ 6211:, 5966:, 5959:, 5944:. 5932:A 5835:, 5828:, 5722:, 5667:. 5659:= 5655:− 5632:+ 5621:+ 5358:. 5354:− 5350:= 5346:+ 5330:+ 5118:+ 5110:= 5102:+ 5094:+ 5090:= 5086:+ 5078:+ 5062:+ 5020:+ 4997:. 3398:, 3389:, 3373:, 3366:, 3214:, 3207:, 3192:, 3167:, 3160:, 2033:, 2029:, 2025:, 2009:, 2005:, 1990:. 1955:AB 1769:, 1694:. 1308:, 1232:. 844:, 822:, 770:, 751:, 747:, 743:, 732:, 720:, 529:, 405:, 401:, 397:, 336:+ 332:= 291:. 248:. 232:, 220:, 216:, 60:, 12923:e 12916:t 12909:v 12896:) 12886:) 12882:( 12850:. 12826:. 12798:. 12772:. 12751:. 12741:. 12722:. 12699:. 12677:. 12651:. 12612:. 12560:. 12528:. 12501:b 12495:a 12489:b 12483:a 12477:r 12446:. 12419:. 12409:) 12407:B 12403:A 12401:( 12346:. 12300:. 12275:. 12244:. 12217:. 12193:. 12171:. 12158:" 11802:v 11794:z 11790:y 11786:x 11782:v 11777:v 11762:1 11746:M 11742:v 11737:v 11732:x 11729:M 11725:x 11720:x 11716:M 11690:) 11684:( 11672:) 11666:( 11661:) 11657:( 11643:. 11617:) 11611:( 11606:) 11602:( 11598:. 11584:. 11555:) 11551:( 11500:. 11497:) 11492:1 11486:x 11475:2 11469:x 11463:( 11455:F 11450:= 11447:W 11416:a 11411:m 11408:= 11403:F 11372:a 11359:t 11355:0 11352:x 11338:, 11333:0 11327:x 11321:+ 11316:v 11311:t 11308:= 11303:t 11297:x 11284:t 11276:v 11266:y 11262:x 11258:x 11254:y 11240:. 11235:3 11229:e 11223:) 11218:3 11214:x 11205:3 11201:y 11197:( 11194:+ 11189:2 11183:e 11177:) 11172:2 11168:x 11159:2 11155:y 11151:( 11148:+ 11143:1 11137:e 11131:) 11126:1 11122:x 11113:1 11109:y 11105:( 11102:= 11097:x 11087:y 11071:3 11068:y 11064:2 11061:y 11057:1 11054:y 11050:y 11046:3 11043:x 11039:2 11036:x 11032:1 11029:x 11025:x 11004:. 10999:3 10993:e 10985:3 10981:x 10977:+ 10972:2 10966:e 10958:2 10954:x 10950:+ 10945:1 10939:e 10931:1 10927:x 10923:= 10918:x 10902:3 10899:x 10895:2 10892:x 10888:1 10885:x 10881:x 10853:t 10851:( 10849:r 10845:r 10841:t 10769:. 10764:2 10759:e 10752:1 10747:e 10742:) 10737:1 10733:b 10727:2 10723:a 10714:2 10710:b 10704:1 10700:a 10696:( 10693:= 10690:) 10685:2 10679:e 10671:2 10667:b 10663:+ 10658:1 10652:e 10644:1 10640:b 10636:( 10630:) 10625:2 10619:e 10611:2 10607:a 10603:+ 10598:1 10592:e 10584:1 10580:a 10576:( 10539:. 10534:4 10528:e 10522:) 10517:4 10513:b 10509:+ 10504:4 10500:a 10496:( 10493:+ 10488:3 10482:e 10476:) 10471:3 10467:b 10463:+ 10458:3 10454:a 10450:( 10447:+ 10438:2 10432:e 10426:) 10421:2 10417:b 10413:+ 10408:2 10404:a 10400:( 10397:+ 10392:1 10386:e 10380:) 10375:1 10371:b 10367:+ 10362:1 10358:a 10354:( 10347:= 10344:) 10339:4 10333:e 10325:4 10321:b 10317:+ 10312:3 10306:e 10298:3 10294:b 10290:+ 10285:2 10279:e 10271:2 10267:b 10263:+ 10258:1 10252:e 10244:1 10240:b 10236:( 10233:+ 10226:) 10221:4 10215:e 10207:4 10203:a 10199:+ 10194:3 10188:e 10180:3 10176:a 10172:+ 10167:2 10161:e 10153:2 10149:a 10145:+ 10140:1 10134:e 10126:1 10122:a 10118:( 10094:, 10089:2 10083:e 10077:) 10072:2 10068:b 10064:+ 10059:2 10055:a 10051:( 10048:+ 10043:1 10037:e 10031:) 10026:1 10022:b 10018:+ 10013:1 10009:a 10005:( 10002:= 9999:) 9994:2 9988:e 9980:2 9976:b 9972:+ 9967:1 9961:e 9953:1 9949:b 9945:( 9942:+ 9939:) 9934:2 9928:e 9920:2 9916:a 9912:+ 9907:1 9901:e 9893:1 9889:a 9885:( 9839:e 9835:n 9831:n 9827:e 9815:n 9811:e 9807:n 9803:e 9795:n 9791:e 9778:c 9774:n 9770:3 9767:n 9763:2 9760:n 9756:1 9753:n 9749:e 9745:3 9742:e 9738:2 9735:e 9731:1 9728:e 9706:3 9701:e 9691:3 9686:n 9681:= 9668:c 9658:2 9653:e 9643:3 9638:n 9633:= 9620:c 9610:1 9605:e 9595:3 9590:n 9585:= 9572:c 9562:3 9557:e 9547:2 9542:n 9537:= 9524:c 9514:2 9509:e 9499:2 9494:n 9489:= 9476:c 9466:1 9461:e 9451:2 9446:n 9441:= 9428:c 9418:3 9413:e 9403:1 9398:n 9393:= 9380:c 9370:2 9365:e 9355:1 9350:n 9345:= 9332:c 9322:1 9317:e 9307:1 9302:n 9297:= 9284:c 9254:k 9250:e 9245:j 9241:n 9228:c 9224:r 9220:q 9216:p 9212:e 9208:w 9204:v 9202:, 9200:u 9196:n 9192:a 9176:. 9171:] 9165:r 9158:q 9151:p 9145:[ 9138:] 9126:c 9114:c 9102:c 9088:c 9076:c 9064:c 9050:c 9038:c 9026:c 9019:[ 9014:= 9009:] 9003:w 8996:v 8989:u 8983:[ 8953:, 8950:r 8941:c 8937:+ 8934:q 8925:c 8921:+ 8918:p 8909:c 8905:= 8898:w 8891:, 8888:r 8879:c 8875:+ 8872:q 8863:c 8859:+ 8856:p 8847:c 8843:= 8836:v 8829:, 8826:r 8817:c 8813:+ 8810:q 8801:c 8797:+ 8794:p 8785:c 8781:= 8774:u 8742:. 8737:3 8732:n 8722:3 8717:e 8712:r 8709:+ 8704:3 8699:n 8689:2 8684:e 8679:q 8676:+ 8671:3 8666:n 8656:1 8651:e 8646:p 8643:= 8636:w 8629:, 8624:2 8619:n 8609:3 8604:e 8599:r 8596:+ 8591:2 8586:n 8576:2 8571:e 8566:q 8563:+ 8558:2 8553:n 8543:1 8538:e 8533:p 8530:= 8523:v 8516:, 8511:1 8506:n 8496:3 8491:e 8486:r 8483:+ 8478:1 8473:n 8463:2 8458:e 8453:q 8450:+ 8445:1 8440:n 8430:1 8425:e 8420:p 8417:= 8410:u 8378:. 8373:3 8368:n 8360:) 8355:3 8350:e 8345:r 8342:+ 8337:2 8332:e 8327:q 8324:+ 8319:1 8314:e 8309:p 8306:( 8303:= 8296:w 8289:, 8284:2 8279:n 8271:) 8266:3 8261:e 8256:r 8253:+ 8248:2 8243:e 8238:q 8235:+ 8230:1 8225:e 8220:p 8217:( 8214:= 8207:v 8200:, 8195:1 8190:n 8182:) 8177:3 8172:e 8167:r 8164:+ 8159:2 8154:e 8149:q 8146:+ 8141:1 8136:e 8131:p 8128:( 8125:= 8118:u 8098:r 8094:q 8090:p 8086:w 8082:v 8078:u 8073:a 8069:w 8065:v 8061:u 8057:r 8053:q 8049:p 8029:. 8024:3 8019:n 8010:a 8006:= 7999:w 7992:, 7987:2 7982:n 7973:a 7969:= 7962:v 7955:, 7950:1 7945:n 7936:a 7932:= 7925:u 7909:n 7891:3 7886:n 7881:w 7878:+ 7873:2 7868:n 7863:v 7860:+ 7855:1 7850:n 7845:u 7842:= 7838:a 7825:a 7821:e 7817:3 7814:n 7810:2 7807:n 7803:1 7800:n 7796:n 7776:. 7771:3 7766:e 7757:a 7753:= 7746:r 7739:, 7734:2 7729:e 7720:a 7716:= 7709:q 7702:, 7697:1 7692:e 7683:a 7679:= 7672:p 7656:e 7640:. 7635:3 7630:e 7625:r 7622:+ 7617:2 7612:e 7607:q 7604:+ 7599:1 7594:e 7589:p 7586:= 7582:a 7569:a 7565:e 7561:3 7558:e 7554:2 7551:e 7547:1 7544:e 7540:e 7519:. 7516:) 7512:a 7504:b 7496:c 7492:( 7486:= 7483:) 7479:c 7471:a 7463:b 7459:( 7453:= 7450:) 7446:b 7438:c 7430:a 7426:( 7420:= 7417:) 7413:a 7405:c 7397:b 7393:( 7390:= 7387:) 7383:b 7375:a 7367:c 7363:( 7360:= 7357:) 7353:c 7345:b 7337:a 7333:( 7309:| 7301:3 7297:c 7289:2 7285:c 7277:1 7273:c 7263:3 7259:b 7251:2 7247:b 7239:1 7235:b 7225:3 7221:a 7213:2 7209:a 7201:1 7197:a 7190:| 7185:= 7182:) 7178:c 7170:b 7162:a 7158:( 7133:c 7129:b 7125:a 7101:. 7098:) 7094:c 7086:b 7082:( 7075:a 7071:= 7068:) 7064:c 7056:b 7048:a 7044:( 7032:c 7029:b 7026:a 6980:. 6975:3 6969:e 6963:) 6958:1 6954:b 6948:2 6944:a 6935:2 6931:b 6925:1 6921:a 6917:( 6914:+ 6909:2 6903:e 6897:) 6892:3 6888:b 6882:1 6878:a 6869:1 6865:b 6859:3 6855:a 6851:( 6848:+ 6843:1 6837:e 6831:) 6826:2 6822:b 6816:3 6812:a 6803:3 6799:b 6793:2 6789:a 6785:( 6782:= 6777:b 6767:a 6752:b 6748:a 6744:b 6740:a 6725:b 6721:a 6717:b 6713:a 6709:b 6705:a 6701:b 6697:a 6682:n 6678:n 6674:b 6670:a 6658:b 6654:a 6646:n 6642:b 6638:a 6634:θ 6617:n 6612:) 6606:( 6592:b 6579:a 6571:= 6567:b 6559:a 6546:b 6542:a 6538:b 6534:a 6491:. 6486:3 6482:b 6476:3 6472:a 6468:+ 6463:2 6459:b 6453:2 6449:a 6445:+ 6440:1 6436:b 6430:1 6426:a 6422:= 6418:b 6410:a 6394:a 6390:b 6386:a 6382:b 6378:a 6370:b 6366:a 6358:θ 6342:, 6325:b 6312:a 6304:= 6300:b 6292:a 6275:a 6261:b 6257:a 6234:a 6230:a 6226:0 6221:a 6217:a 6213:0 6193:0 6147:3 6142:e 6130:a 6120:3 6116:a 6110:+ 6105:2 6100:e 6088:a 6078:2 6074:a 6068:+ 6063:1 6058:e 6046:a 6036:1 6032:a 6026:= 6016:a 6007:a 6001:= 5992:a 5977:a 5973:) 5971:3 5968:a 5964:2 5961:a 5957:1 5954:a 5950:a 5942:â 5921:â 5917:a 5892:. 5886:a 5878:a 5872:= 5864:a 5840:3 5837:e 5833:2 5830:e 5826:1 5823:e 5803:, 5796:2 5791:3 5787:a 5783:+ 5778:2 5773:2 5769:a 5765:+ 5760:2 5755:1 5751:a 5745:= 5737:a 5714:a 5703:a 5699:a 5695:a 5665:b 5661:a 5657:b 5653:a 5649:r 5645:b 5641:a 5637:b 5634:r 5630:a 5627:r 5623:b 5619:a 5617:( 5615:r 5604:a 5600:a 5596:a 5585:r 5581:r 5577:r 5570:r 5566:r 5562:r 5558:r 5542:. 5537:3 5532:e 5527:) 5522:3 5518:a 5514:r 5511:( 5508:+ 5503:2 5498:e 5493:) 5488:2 5484:a 5480:r 5477:( 5474:+ 5469:1 5464:e 5459:) 5454:1 5450:a 5446:r 5443:( 5440:= 5436:a 5432:r 5404:r 5371:b 5367:a 5356:b 5352:a 5348:a 5340:b 5332:a 5324:a 5320:b 5316:b 5312:a 5308:a 5304:b 5288:. 5283:3 5278:e 5273:) 5268:3 5264:b 5255:3 5251:a 5247:( 5244:+ 5239:2 5234:e 5229:) 5224:2 5220:b 5211:2 5207:a 5203:( 5200:+ 5195:1 5190:e 5185:) 5180:1 5176:b 5167:1 5163:a 5159:( 5156:= 5152:b 5144:a 5131:b 5127:a 5120:c 5116:b 5112:a 5108:c 5104:b 5100:a 5096:a 5092:b 5088:b 5084:a 5080:b 5076:a 5072:b 5068:a 5064:b 5060:a 5052:b 5048:a 5037:b 5033:a 5022:b 5018:a 5014:b 5010:a 5006:a 5002:b 4995:b 4991:a 4973:. 4968:3 4963:e 4958:) 4953:3 4949:b 4945:+ 4940:3 4936:a 4932:( 4929:+ 4924:2 4919:e 4914:) 4909:2 4905:b 4901:+ 4896:2 4892:a 4888:( 4885:+ 4880:1 4875:e 4870:) 4865:1 4861:b 4857:+ 4852:1 4848:a 4844:( 4841:= 4837:b 4833:+ 4829:a 4818:b 4814:a 4769:. 4764:3 4760:b 4753:= 4748:3 4744:a 4739:, 4734:2 4730:b 4723:= 4718:2 4714:a 4709:, 4704:1 4700:b 4693:= 4688:1 4684:a 4657:3 4651:e 4643:3 4639:b 4635:+ 4630:2 4624:e 4616:2 4612:b 4608:+ 4603:1 4597:e 4589:1 4585:b 4581:= 4576:b 4548:3 4542:e 4534:3 4530:a 4526:+ 4521:2 4515:e 4507:2 4503:a 4499:+ 4494:1 4488:e 4480:1 4476:a 4472:= 4467:a 4427:. 4422:3 4418:b 4414:= 4409:3 4405:a 4400:, 4395:2 4391:b 4387:= 4382:2 4378:a 4373:, 4368:1 4364:b 4360:= 4355:1 4351:a 4328:3 4322:e 4314:3 4310:b 4306:+ 4301:2 4295:e 4287:2 4283:b 4279:+ 4274:1 4268:e 4260:1 4256:b 4252:= 4247:b 4223:3 4217:e 4209:3 4205:a 4201:+ 4196:2 4190:e 4182:2 4178:a 4174:+ 4169:1 4163:e 4155:1 4151:a 4147:= 4142:a 4113:. 4108:3 4102:e 4094:3 4090:a 4086:+ 4081:2 4075:e 4067:2 4063:a 4059:+ 4054:1 4048:e 4040:1 4036:a 4032:= 4027:a 4015:a 4001:) 3998:1 3995:, 3992:0 3989:, 3986:0 3983:( 3980:= 3975:3 3969:e 3960:, 3957:) 3954:0 3951:, 3948:1 3945:, 3942:0 3939:( 3936:= 3931:2 3925:e 3916:, 3913:) 3910:0 3907:, 3904:0 3901:, 3898:1 3895:( 3892:= 3887:1 3881:e 3771:, 3756:, 3747:r 3736:( 3714:z 3708:, 3693:, 3673:( 3647:z 3641:, 3632:y 3626:, 3617:x 3547:i 3543:e 3527:. 3522:k 3515:z 3511:a 3507:+ 3502:j 3495:y 3491:a 3487:+ 3482:i 3475:x 3471:a 3467:= 3462:z 3457:a 3452:+ 3447:y 3442:a 3437:+ 3432:x 3427:a 3422:= 3418:a 3404:z 3400:a 3395:y 3391:a 3386:x 3382:a 3377:z 3375:a 3370:y 3368:a 3363:x 3361:a 3307:z 3301:, 3292:y 3286:, 3277:x 3253:k 3249:, 3245:j 3241:, 3237:i 3219:3 3216:a 3212:2 3209:a 3205:1 3202:a 3198:z 3194:y 3190:x 3186:a 3172:3 3169:a 3165:2 3162:a 3158:1 3155:a 3139:, 3134:3 3128:e 3120:3 3116:a 3112:+ 3107:2 3101:e 3093:2 3089:a 3085:+ 3080:1 3074:e 3066:1 3062:a 3058:= 3053:3 3048:a 3043:+ 3038:2 3033:a 3028:+ 3023:1 3018:a 3013:= 3009:a 2983:, 2980:) 2977:1 2974:, 2971:0 2968:, 2965:0 2962:( 2957:3 2953:a 2949:+ 2946:) 2943:0 2940:, 2937:1 2934:, 2931:0 2928:( 2923:2 2919:a 2915:+ 2912:) 2909:0 2906:, 2903:0 2900:, 2897:1 2894:( 2889:1 2885:a 2881:= 2878:) 2873:3 2869:a 2865:, 2860:2 2856:a 2852:, 2847:1 2843:a 2839:( 2836:= 2832:a 2820:R 2815:a 2807:z 2803:y 2799:x 2785:. 2782:) 2779:1 2776:, 2773:0 2770:, 2767:0 2764:( 2761:= 2756:3 2750:e 2741:, 2738:) 2735:0 2732:, 2729:1 2726:, 2723:0 2720:( 2717:= 2712:2 2706:e 2697:, 2694:) 2691:0 2688:, 2685:0 2682:, 2679:1 2676:( 2673:= 2668:1 2662:e 2645:n 2629:. 2624:T 2620:] 2614:3 2610:a 2601:2 2597:a 2588:1 2584:a 2580:[ 2577:= 2572:] 2564:3 2560:a 2550:2 2546:a 2536:1 2532:a 2525:[ 2520:= 2516:a 2481:. 2478:) 2473:n 2469:a 2465:, 2460:1 2454:n 2450:a 2446:, 2440:, 2435:3 2431:a 2427:, 2422:2 2418:a 2414:, 2409:1 2405:a 2401:( 2398:= 2394:a 2382:R 2361:. 2358:) 2353:z 2349:a 2345:, 2340:y 2336:a 2332:, 2327:x 2323:a 2319:( 2316:= 2312:a 2291:. 2288:) 2283:3 2279:a 2275:, 2270:2 2266:a 2262:, 2257:1 2253:a 2249:( 2246:= 2242:a 2230:R 2202:A 2199:O 2174:. 2171:) 2168:3 2165:, 2162:2 2159:( 2156:= 2152:a 2141:A 2137:O 2110:- 2108:n 2104:n 2092:n 2077:A 2019:B 1999:A 1976:a 1931:B 1928:A 1914:B 1910:A 1877:a 1860:a 1839:a 1817:a 1801:w 1778:v 1755:u 1737:B 1733:A 1680:K 1614:n 1606:n 1552:. 1548:) 1545:7 1542:, 1539:2 1536:, 1533:1 1527:( 1524:= 1521:) 1518:4 1515:+ 1512:3 1509:, 1506:0 1503:+ 1500:2 1497:, 1494:2 1488:1 1485:( 1482:= 1479:) 1476:4 1473:, 1470:0 1467:, 1464:2 1458:( 1455:+ 1452:) 1449:3 1446:, 1443:2 1440:, 1437:1 1434:( 1420:x 1414:O 1406:y 1400:y 1395:x 1389:x 1368:B 1365:A 1350:B 1343:A 1294:y 1266:s 1262:F 1258:F 1246:F 1224:n 1209:, 1204:n 1199:R 1177:. 1168:P 1165:O 1152:P 1132:O 1118:. 1113:n 1108:R 1092:n 1088:n 1065:n 1060:R 1033:n 1028:R 1000:n 995:R 957:E 923:, 915:E 885:, 877:E 861:E 834:B 830:A 826:) 824:B 820:A 818:( 799:. 790:B 787:A 774:) 772:B 768:A 766:( 753:C 749:D 745:B 741:A 736:) 734:D 730:C 728:( 724:) 722:B 718:A 716:( 637:B 629:A 594:B 591:A 377:v 347:s 338:v 334:s 330:q 202:B 198:A 185:B 181:A 165:. 153:B 150:A 136:B 129:A 51:B 47:A 38:. 20:)

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Index