4962:
521:
105:
1625:
30:
5226:
53:
2165:
2933:
529:
87:
954:
511:
meeting of the next
Congress, which was to have taken place in Stockholm in 1916, but which was omitted because of the war. The committee on units and symbols met a similar fate. It published in 1921 a proposed notation for vector quantities, which aroused at once and from many sides the most violent opposition.
1145:
497:
The terms line-segment, plane-segment, plane magnitude, inner and outer product come from
Grassmann, while the words scalar, vector, scalar product, and vector product came from Hamilton. The disciples of Grassmann, in other ways so orthodox, replaced in part the appropriate expressions of the master
3650:
is represented in the same manners as algebraic multiplication. A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they
510:
The
Committee which was set up in Rome for the unification of vector notation did not have the slightest success, as was to have been expected. At the following Congress in Cambridge (1912), they had to explain that they had not finished their task, and to request that their time be extended to the
498:
by others. The existing terminologies were merged or modified, and the symbols which indicate the separate operations have been used with the greatest arbitrariness. On these accounts even for the expert, a great lack of clearness has crept into this field, which is mathematically so simple.
4409:
1581:
785:
3207:
3272:
960:
4331:
2484:
1754:
Vectors can be specified using either ordered pair notation (a subset of ordered set notation using only two components), or matrix notation, as with rectangular coordinates. In these forms, the first component of the vector is
3012:
Spherical vectors are specified like polar vectors, where the zenith angle is concatenated as a third component to form ordered triplets and matrices. The azimuth and zenith angles may be both prefixed with the angle symbol
2546:
1945:
3146:
2000:
1489:
3730:
is performed by adding the scalar multiple of −1 with the second vector operand to the first vector operand. This can be represented by the use of the minus sign as an operator. The difference between two vectors
3391:
3462:
3093:
2426:
667:
1892:
4056:
1406:
1353:
2858:
2730:
4014:
of two vectors (also known as the scalar product, not to be confused with scalar multiplication) is represented as an ordered pair enclosed in angle brackets. The inner product of two vectors
2922:
2376:
1459:
4255:
4219:
2794:
1848:
4134:
1744:
949:{\displaystyle \mathbf {v} ={\begin{bmatrix}v_{1}&v_{2}&\cdots &v_{n-1}&v_{n}\end{bmatrix}}={\begin{pmatrix}v_{1}&v_{2}&\cdots &v_{n-1}&v_{n}\end{pmatrix}}}
2105:
3773:
3804:
3638:
2154:
3857:
3887:
2273:
1705:
699:
4098:. In addition to the standard inner product notation, the dot product notation (using the dot as an operator) can also be used (and is more common). The dot product of two vectors
3918:
3714:
1484:
4522:
4181:
4092:
3962:
3724:
Using the algebraic properties of subtraction and division, along with scalar multiplication, it is also possible to “subtract” two vectors and “divide” a vector by a scalar.
3585:
3556:
3522:
1297:
1255:
1226:
736:
4483:
3996:
3685:
2049:
explicitly. This can be unwieldy, but is useful for avoiding the confusion with two-dimensional rectangular vectors that arises from using ordered pair or matrix notation.
777:
261:
4444:
3152:
203:
483:
their components are printed in the same Gothic types. The more usual way of making a typographical distinction between the two has been adopted for this translation."
3031:
2605:
2317:
2031:
1795:
3213:
1140:{\displaystyle \mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n-1}\\v_{n}\end{bmatrix}}={\begin{pmatrix}v_{1}\\v_{2}\\\vdots \\v_{n-1}\\v_{n}\end{pmatrix}}}
4577:
4301:
2432:
2299:
as a third component to form ordered triplets (again, a subset of ordered set notation) and matrices. The angle may be prefixed with the angle symbol (
3033:); the prefix should be used consistently to produce the distance-angle-angle combination that distinguishes spherical vectors from cylindrical ones.
2490:
1898:
3099:
4404:{\displaystyle \mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}}}
1951:
4223:
By some conventions (e.g. in France and in some areas of higher mathematics), this is also denoted by a wedge, which avoids confusion with the
4025:
167:
3326:
2319:); the distance-angle-distance combination distinguishes cylindrical vectors in this notation from spherical vectors in similar notation.
5255:
3816:
of the scalar operand. This can be represented by the use of the fraction bar or division signs as operators. The quotient of a vector
3397:
4138:
In some older literature, the dot product is implied between two vectors written side-by-side. This notation can be confused with the
3046:
2976:
A spherical vector is another method for extending the concept of polar vectors into three dimensions. It is akin to an arrow in the
4230:
4194:
2382:
580:
4109:
1266:
below is zero) can be specified as the sum of the scalar multiples of the components of the vector with the members of the standard
4820:
1854:
1188:
503:
2980:. A spherical vector is specified by a magnitude, an azimuth angle, and a zenith angle. The magnitude is usually represented as
1576:{\displaystyle \mathbf {v} =v_{x}{\boldsymbol {\hat {\imath }}}+v_{y}{\boldsymbol {\hat {\jmath }}}+v_{z}{\boldsymbol {\hat {k}}}}
489:
commented on differences in notation of vectors and their operations in 1925 through a Mr. Seyfarth who prepared a supplement to
5153:
5211:
3613:
3297:
Like polar and cylindrical vectors, spherical vectors can be specified using simplified autonomous equations, in this case for
2196:
A cylindrical vector is an extension of the concept of polar coordinates into three dimensions. It is akin to an arrow in the
1358:
1305:
2800:
1636:. In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°).
2672:
2864:
2332:
3939:
1411:
4787:
2736:
1810:
1777:). To differentiate polar coordinates from rectangular coordinates, the angle may be prefixed with the angle symbol,
1668:), is the angle, usually measured counterclockwise, between a fixed direction, typically that of the positive
5201:
4654:
1710:
138:
2062:
457:. But German mathematicians were not taken with quaternions as much as were English-speaking mathematicians. When
5163:
5099:
475:
3745:
5250:
3779:
2111:
404:
1672:-axis, and the direction from the origin to the point. The angle is typically reduced to lie within the range
3830:
2197:
3863:
3599:
2243:
1675:
1616:. Scalar components may be positive or negative; the absolute value of a scalar component is its magnitude.
678:
4941:
4813:
3651:
risk confusion with dot products and cross products, which operate on two vectors. The product of a scalar
4682:
3893:
5046:
4896:
3691:
2977:
543:
4951:
4845:
2615:
A cylindrical vector can also be specified directly, using simplified autonomous equations that define
1467:
4498:
4270:
4157:
4068:
3561:
3532:
3498:
1273:
1231:
1202:
712:
5191:
4840:
4459:
3969:
3202:{\displaystyle \mathbf {v} ={\begin{bmatrix}\rho &\angle \theta &\angle \phi \end{bmatrix}}}
479:
by
Sommerfeld, vector notation was the subject of a footnote: "In the original German text, vectors
5183:
5066:
4752:
3665:
3602:
is represented with the plus sign used as an operator between two vectors. The sum of two vectors
4777:
760:
234:
5229:
5158:
4936:
4806:
4422:
1645:
1644:
of a point in a plane may be considered as a two dimensional vector. Such a vector consists of a
462:
228:
179:
4993:
4926:
4916:
3932:
of a vector is represented with double bars on both sides of the vector. The norm of a vector
3813:
1267:
206:
4641:
4637:
4561:
3473:
3267:{\displaystyle \mathbf {v} ={\begin{bmatrix}\rho \\\angle \theta \\\angle \phi \end{bmatrix}}}
380:
severed the two products to make the quaternion operation useful for students in his textbook
5008:
5003:
4998:
4931:
4876:
4713:
3647:
3495:
3040:
can be represented as any of the following, using either ordered triplet or matrix notation:
3016:
2590:
2326:
can be represented as any of the following, using either ordered triplet or matrix notation:
2302:
2016:
1780:
567:
537:
282:, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion
146:
134:
3000:-axis. Both angles are typically reduced to lie within the range from zero (inclusive) to 2
5018:
4983:
4970:
4861:
4678:
4610:
742:
containing the ordered set of components. A vector specified as a row matrix is known as a
739:
443:
416:
391:
205:. In advanced mathematics, vectors are often represented in a simple italic type, like any
4183:) is represented using the rotated cross as an operator. The cross product of two vectors
1804:
can be represented as any of the following, using either ordered pair or matrix notation:
8:
5196:
5076:
5051:
4901:
4537:
4011:
3491:
3483:
428:
382:
4683:
A System of
Notation for Vector-Analysis; with a Discussion of the Underlying Principles
4581:
2240:-axis in the positive direction; the angle is typically reduced to lie within the range
434:
In 1912, J.B. Shaw contributed his "Comparative
Notation for Vector Expressions" to the
4906:
4662:
4492:
4259:
In some older literature, the following notation is used for the cross product between
3929:
3487:
439:
5104:
5061:
4988:
4881:
4783:
2479:{\displaystyle \mathbf {v} ={\begin{bmatrix}r&\angle \theta &h\end{bmatrix}}}
1653:
1641:
547:
466:
450:
264:
224:
5109:
5013:
4866:
4629:
4618:
4532:
1187:. In some advanced contexts, a row and a column vector have different meaning; see
412:
142:
58:
2295:
Cylindrical vectors use polar coordinates, where the second distance component is
5168:
4961:
4921:
4911:
2639:). Consistency should be used when choosing the names to use for the variables;
454:
446:
described 15 criteria for clear expression with vectors in the same publication.
387:
4701:
2041:
Vectors can also be specified using simplified autonomous equations that define
1648:(or length) and a direction (or angle). The magnitude, typically represented as
5173:
5094:
4829:
4139:
3999:
2541:{\displaystyle \mathbf {v} ={\begin{bmatrix}r\\\angle \theta \\h\end{bmatrix}}}
1940:{\displaystyle \mathbf {v} ={\begin{bmatrix}r&\angle \theta \end{bmatrix}}}
1661:
673:
470:
399:
395:
377:
275:
110:
92:
3312:
A three-dimensional vector whose magnitude is 5 units, whose azimuth angle is
3141:{\displaystyle \mathbf {v} =\langle \rho ,\angle \theta ,\angle \phi \rangle }
5244:
5206:
5129:
5089:
5056:
5036:
4224:
4151:
3588:
2296:
747:
420:
2052:
A two-dimensional vector whose magnitude is 5 units, and whose direction is
1995:{\displaystyle \mathbf {v} ={\begin{bmatrix}r\\\angle \theta \end{bmatrix}}}
520:
96:
Two equal-length sequences of coordinate vectors and returns a single number
5139:
5028:
4978:
4871:
4718:
4315:
3482:, the operations of vector addition and scalar multiplication are defined.
3479:
150:
4597:
5119:
5084:
5041:
4886:
4734:
4697:
4666:
4563:
Principles and
Applications of Mathematics for Communications-electronics
4095:
3526:
1656:, to the point which is represented. The angle, typically represented as
1300:
486:
458:
122:
75:
4614:
2056:/9 radians (20°), can be specified using either of the following forms:
502:
Efforts to unify the various notational terms through committees of the
5148:
4891:
4453:
3386:{\displaystyle \rho =5,\ \theta ={\pi \over 9},\ \phi ={\pi \over 4}}
2650:
A three-dimensional vector, the magnitude of whose projection onto the
743:
473:
published G. Kuerti’s translation of the second edition of volume 2 of
279:
157:
104:
1624:
4946:
29:
4702:
Die Rolle Arnold
Sommerfeld bei der Diskussion um die Vektorrechnung
2164:
1486:
can be specified in the following form, using unit vector notation:
52:
5114:
4416:
4321:
2666:-plane is 3 units can be specified in any of the following forms:
779:
can be specified in either of the following forms using matrices:
215:
include
Cartesian, polar, cylindrical, and spherical coordinates.
4798:
3457:{\displaystyle \rho =5,\ \theta =20^{\circ },\ \phi =45^{\circ }}
2932:
424:
126:
3088:{\displaystyle \mathbf {v} =(\rho ,\angle \theta ,\angle \phi )}
528:
114:
The cross-product in respect to a right-handed coordinate system
5124:
4542:
4227:
since the two are functionally equivalent in three dimensions:
2421:{\displaystyle \mathbf {v} =\langle r,\angle \theta ,h\rangle }
662:{\displaystyle \mathbf {v} =(v_{1},v_{2},\dots ,v_{n-1},v_{n})}
4600:. International Organization for Standardization. August 2019.
3966:
The norm is also sometimes represented with single bars, like
2236:), is measured as the offset from the line collinear with the
2208:-plane (a height). The first distance, usually represented as
419:
to distinguish vectors from scalars. He criticized the use of
86:
4739:
Elementary
Mathematics from an Advanced Standpoint – Geometry
4598:"ISO 80000-2:2019 Quantities and units — Part 2: Mathematics"
2220:), is the magnitude of the projection of the vector onto the
1887:{\displaystyle \mathbf {v} =\langle r,\angle \theta \rangle }
1665:
571:
491:
Elementary Mathematics from an Advanced Standpoint — Geometry
176:, or non-bold italic serif accented by a right arrow, as in
2988:, is the (counterclockwise) offset from the positive
4311:
4051:{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle }
3719:
2233:
2217:
2200:. A cylindrical vector is specified by a distance in the
3812:
is performed by multiplying the vector operand with the
3739:
can be represented in either of the following fashions:
3494:
also define an operation known as the inner product. In
3230:
3169:
2507:
2449:
1968:
1915:
1401:{\displaystyle {\boldsymbol {\hat {\jmath }}}=(0,1,0)}
1348:{\displaystyle {\boldsymbol {\hat {\imath }}}=(1,0,0)}
1061:
977:
878:
802:
4737:, translators E.R. Hendrick & C.A. Noble (1939)
4615:
Vector Analysis, based on the Lectures of J. W. Gibbs
4501:
4462:
4425:
4334:
4273:
4233:
4197:
4160:
4112:
4071:
4028:
3972:
3942:
3896:
3866:
3833:
3782:
3748:
3694:
3668:
3659:
can be represented in any of the following fashions:
3616:
3564:
3535:
3501:
3400:
3329:
3216:
3155:
3102:
3049:
3019:
3007:
2972:
have been swapped compared to the physics convention.
2867:
2803:
2739:
2675:
2593:
2493:
2435:
2385:
2335:
2305:
2290:
2246:
2114:
2065:
2019:
1954:
1901:
1857:
1813:
1783:
1713:
1678:
1492:
1470:
1414:
1361:
1308:
1276:
1234:
1205:
963:
788:
763:
715:
681:
583:
237:
182:
2853:{\displaystyle \rho =5,\ \phi ={\pi \over 9},\ z=3}
1749:
4516:
4477:
4438:
4403:
4295:
4249:
4213:
4175:
4128:
4086:
4050:
3990:
3956:
3912:
3881:
3851:
3824:can be represented in any of the following forms:
3798:
3767:
3708:
3679:
3632:
3579:
3550:
3516:
3456:
3385:
3266:
3201:
3140:
3087:
3025:
2916:
2852:
2788:
2725:{\displaystyle r=5,\ \theta ={\pi \over 9},\ h=3}
2724:
2599:
2540:
2478:
2420:
2370:
2311:
2267:
2148:
2099:
2025:
1994:
1939:
1886:
1842:
1789:
1738:
1699:
1575:
1478:
1453:
1400:
1347:
1291:
1249:
1220:
1139:
948:
771:
730:
693:
661:
255:
197:
2992:-axis. The zenith angle, usually represented as
2917:{\displaystyle \rho =5,\ \phi =20^{\circ },\ z=3}
2654:-plane is 5 units, whose angle from the positive
2371:{\displaystyle \mathbf {v} =(r,\angle \theta ,h)}
1567:
1542:
1517:
1421:
1368:
1315:
746:; one specified as a column matrix is known as a
263:which resulted in the concept of a vector as an
170:(ISO) recommends either bold italic serif, as in
5242:
1628:Points in the polar coordinate system with pole
1454:{\displaystyle {\boldsymbol {\hat {k}}}=(0,0,1)}
4250:{\displaystyle \mathbf {u} \wedge \mathbf {v} }
4214:{\displaystyle \mathbf {u} \times \mathbf {v} }
2789:{\displaystyle r=5,\ \theta =20^{\circ },\ h=3}
1843:{\displaystyle \mathbf {v} =(r,\angle \theta )}
4129:{\displaystyle \mathbf {u} \cdot \mathbf {v} }
2275:. The second distance, usually represented as
469:to standardize vector notation. In 1950, when
168:International Organization for Standardization
4814:
2984:. The azimuth angle, usually represented as
1739:{\displaystyle 0\leq \theta <360^{\circ }}
1652:, is the distance from a starting point, the
546:, a vector may be specified by its Cartesian
4782:. Dover Publications. p. 134 (Vol. 2).
4045:
4029:
3951:
3943:
3316:/9 radians (20°), and whose zenith angle is
3135:
3111:
2662:/9 radians (20°), and whose height from the
2415:
2394:
2180:. The dot is the point with radial distance
2168:A cylindrical coordinate system with origin
2100:{\displaystyle r=5,\ \theta ={\pi \over 9}}
1881:
1866:
688:
682:
160:is lower case, upright boldface type, as in
4655:Comparative Notation for Vector Expressions
2224:-plane. The angle, usually represented as
336:k, the vector part. Using the modern terms
16:Use of coordinates for representing vectors
4821:
4807:
3768:{\displaystyle \mathbf {u} +-\mathbf {v} }
2204:-plane, an angle, and a distance from the
515:
4432:
4163:
4094:, the inner product is also known as the
4074:
3799:{\displaystyle \mathbf {u} -\mathbf {v} }
3633:{\displaystyle \mathbf {u} +\mathbf {v} }
3567:
3538:
3504:
2149:{\displaystyle r=5,\ \theta =20^{\circ }}
1279:
1237:
1208:
738:can also be specified as a row or column
718:
3642:
2931:
2163:
1623:
1189:covariance and contravariance of vectors
527:
519:
504:International Congress of Mathematicians
3852:{\displaystyle {1 \over c}\mathbf {v} }
3587:, an additional operation known as the
2322:A three-dimensional cylindrical vector
1564:
1539:
1514:
1472:
1418:
1365:
1312:
1194:
493:after "repeated conferences" with him.
5243:
5212:Comparison of linear algebra libraries
4775:
4730:
4728:
4726:
4576:
3882:{\displaystyle {\mathbf {v} \over c}}
3720:Vector subtraction and scalar division
3486:also define an operation known as the
3320:/4 radians (45°) can be specified as:
2555:is the magnitude of the projection of
2287:-plane to the endpoint of the vector.
2268:{\displaystyle 0\leq \theta <2\pi }
2159:
1800:Two-dimensional polar coordinates for
1700:{\displaystyle 0\leq \theta <2\pi }
694:{\displaystyle \langle \dots \rangle }
4802:
3036:A three-dimensional spherical vector
1299:. The basis is represented with the
4538:ISO 31-11 § Vectors and tensors
3913:{\displaystyle {\mathbf {v} \div c}}
3525:, the inner product is known as the
3292:
2927:
2610:
2013:is the angle, and the angle symbol (
1619:
4779:A History of Mathematical Notations
4723:
3709:{\displaystyle k\cdot \mathbf {v} }
2184: = 4, angular coordinate
13:
4828:
4750:
4687:Bulletin of the Quaternion Society
4502:
4463:
4426:
4392:
4388:
4369:
4365:
4346:
4342:
3594:
3474:Euclidean vector § Operations
3250:
3240:
3185:
3177:
3129:
3120:
3076:
3067:
3020:
2996:, is the offset from the positive
2594:
2567:is the angle between the positive
2517:
2457:
2403:
2353:
2306:
2036:
2020:
1978:
1923:
1875:
1831:
1784:
704:
453:in 1841, and again in 1862 in the
302:k, Hamilton used two projections:
156:For denoting a vector, the common
14:
5267:
5256:Vectors (mathematics and physics)
3490:(or determination of magnitude).
1479:{\displaystyle {\boldsymbol {v}}}
701:are used instead of parentheses.
553:
5225:
5224:
5202:Basic Linear Algebra Subprograms
4960:
4517:{\displaystyle \nabla \times F.}
4382:
4359:
4336:
4286:
4278:
4243:
4235:
4207:
4199:
4176:{\displaystyle \mathbb {R} ^{3}}
4145:
4122:
4114:
4087:{\displaystyle \mathbb {R} ^{n}}
4041:
4033:
4005:
3998:, but this can be confused with
3979:
3957:{\displaystyle \|\mathbf {v} \|}
3947:
3899:
3870:
3845:
3792:
3784:
3761:
3750:
3702:
3673:
3626:
3618:
3580:{\displaystyle \mathbb {R} ^{7}}
3551:{\displaystyle \mathbb {R} ^{3}}
3517:{\displaystyle \mathbb {R} ^{n}}
3218:
3157:
3104:
3051:
3008:Ordered set and matrix notations
2495:
2437:
2387:
2337:
2291:Ordered set and matrix notations
1956:
1903:
1859:
1815:
1750:Ordered set and matrix notations
1494:
1292:{\displaystyle \mathbb {R} ^{3}}
1250:{\displaystyle \mathbb {R} ^{2}}
1221:{\displaystyle \mathbb {R} ^{3}}
965:
790:
765:
731:{\displaystyle \mathbb {R} ^{n}}
585:
463:German mathematical encyclopedia
103:
85:
51:
28:
5100:Seven-dimensional cross product
4769:
4744:
4636:, Volume 28. James Gray, 1891.
4478:{\displaystyle \nabla \cdot F,}
1766:), and the second component is
574:(ordered list) of coordinates:
476:Lectures on Theoretical Physics
4714:Mechanics of Deformable Bodies
4706:
4691:
4672:
4647:
4623:
4604:
4590:
4570:
4554:
4290:
4274:
4060:
3991:{\displaystyle |\mathbf {v} |}
3984:
3974:
3082:
3058:
2365:
2344:
2188: = 130°, and height
1837:
1822:
1448:
1430:
1395:
1377:
1342:
1324:
1228:(or fewer dimensions, such as
656:
592:
449:Vector ideas were advanced by
189:
1:
4548:
4320:Vector notation is used with
3680:{\displaystyle k\mathbf {v} }
3467:
2198:cylindrical coordinate system
1612:are the scalar components of
4942:Eigenvalues and eigenvectors
2587:. Again, the angle symbol (
772:{\displaystyle \mathbf {v} }
278:around 1843, as he revealed
256:{\displaystyle AB\bumpeq CD}
7:
4526:
4439:{\displaystyle \nabla f\,,}
4002:(which is a type of norm).
2978:spherical coordinate system
2283:, is the distance from the
1464:A three-dimensional vector
544:Cartesian coordinate system
506:were described as follows:
62:Describing an arrow vector
10:
5272:
4753:"Precalculus 6-03 Vectors"
4309:
3471:
3285:is the azimuth angle, and
2583:-plane to the endpoint of
535:
402:, which was introduced in
394:supplied notation for the
218:
198:{\displaystyle {\vec {v}}}
5220:
5182:
5138:
5075:
5027:
4969:
4958:
4854:
4836:
4191:would be represented as:
4022:would be represented as:
3610:would be represented as:
2643:should not be mixed with
570:can be specified using a
313:, for the scalar part of
4776:Cajori, Florian (2011).
4487:and with a vector field
4305:
2176:, and longitudinal axis
4411:With a scalar function
4106:can be represented as:
3936:can be represented as:
3923:
3026:{\displaystyle \angle }
2936:Spherical coordinates (
2600:{\displaystyle \angle }
2579:is the height from the
2312:{\displaystyle \angle }
2026:{\displaystyle \angle }
1790:{\displaystyle \angle }
516:Rectangular coordinates
231:directed line segments
227:introduced the idea of
4927:Row and column vectors
4634:The Electrical Journal
4586:. J. Wiley & sons.
4518:
4479:
4440:
4405:
4297:
4251:
4215:
4177:
4130:
4088:
4052:
3992:
3958:
3914:
3883:
3853:
3814:multiplicative inverse
3800:
3769:
3710:
3681:
3634:
3581:
3552:
3518:
3458:
3387:
3268:
3203:
3142:
3089:
3027:
2973:
2918:
2854:
2790:
2726:
2601:
2542:
2480:
2422:
2372:
2313:
2269:
2193:
2150:
2101:
2027:
1996:
1941:
1888:
1844:
1791:
1740:
1701:
1637:
1577:
1480:
1455:
1402:
1349:
1293:
1251:
1222:
1183:are the components of
1141:
950:
773:
732:
695:
663:
533:
525:
513:
500:
257:
213:Vector representations
199:
158:typographic convention
5251:Mathematical notation
4932:Row and column spaces
4877:Scalar multiplication
4578:Coffin, Joseph George
4519:
4480:
4441:
4406:
4298:
4252:
4216:
4178:
4142:between two vectors.
4131:
4089:
4053:
3993:
3959:
3915:
3884:
3854:
3801:
3770:
3711:
3682:
3648:Scalar multiplication
3643:Scalar multiplication
3635:
3582:
3553:
3519:
3472:Further information:
3459:
3388:
3289:is the zenith angle.
3269:
3204:
3143:
3090:
3028:
2935:
2919:
2855:
2791:
2727:
2602:
2543:
2481:
2423:
2373:
2314:
2270:
2167:
2151:
2102:
2028:
1997:
1942:
1889:
1845:
1792:
1741:
1702:
1627:
1578:
1481:
1456:
1403:
1350:
1294:
1252:
1223:
1142:
951:
774:
733:
696:
664:
568:real coordinate space
538:Real coordinate space
531:
523:
508:
495:
258:
200:
145:, or more generally,
5067:Gram–Schmidt process
5019:Gaussian elimination
4679:Alexander Macfarlane
4611:Edwin Bidwell Wilson
4566:. 1992. p. 123.
4499:
4460:
4448:with a vector field
4423:
4332:
4271:
4231:
4195:
4158:
4110:
4069:
4026:
3970:
3940:
3894:
3864:
3831:
3780:
3746:
3692:
3666:
3614:
3562:
3533:
3499:
3492:Inner product spaces
3484:Normed vector spaces
3398:
3327:
3214:
3153:
3100:
3047:
3017:
2865:
2801:
2737:
2673:
2591:
2491:
2433:
2383:
2333:
2303:
2244:
2112:
2063:
2017:
1952:
1899:
1855:
1811:
1781:
1711:
1676:
1490:
1468:
1412:
1359:
1306:
1274:
1232:
1203:
1195:Unit vector notation
961:
786:
761:
757:-dimensional vector
713:
679:
581:
444:Alexander Macfarlane
392:Josiah Willard Gibbs
235:
180:
5197:Numerical stability
5077:Multilinear algebra
5052:Inner product space
4902:Linear independence
4154:of two vectors (in
2948:) as often used in
2160:Cylindrical vectors
461:was organizing the
383:Elements of Dynamic
267:of such segments.
133:is a commonly used
66:by its coordinates
4907:Linear combination
4663:Quaternion Society
4514:
4475:
4436:
4401:
4293:
4247:
4211:
4173:
4126:
4084:
4048:
3988:
3954:
3910:
3879:
3849:
3796:
3765:
3728:Vector subtraction
3706:
3677:
3630:
3577:
3548:
3514:
3454:
3383:
3281:is the magnitude,
3264:
3258:
3199:
3193:
3138:
3085:
3023:
2974:
2964:. The meanings of
2960:, and polar angle
2956:, azimuthal angle
2952:: radial distance
2914:
2850:
2786:
2722:
2597:
2538:
2532:
2476:
2470:
2418:
2368:
2309:
2265:
2232:(the Greek letter
2216:(the Greek letter
2194:
2146:
2097:
2023:
2009:is the magnitude,
1992:
1986:
1937:
1931:
1884:
1840:
1787:
1736:
1697:
1638:
1573:
1476:
1451:
1398:
1345:
1289:
1247:
1218:
1137:
1131:
1047:
946:
940:
864:
769:
728:
691:
659:
534:
532:Rectangular cuboid
526:
440:Quaternion Society
346:quaternion product
253:
195:
137:for representing
5238:
5237:
5105:Geometric algebra
5062:Kronecker product
4897:Linear projection
4882:Vector projection
4751:Wright, Richard.
4653:J.B. Shaw (1912)
4399:
4376:
4353:
3877:
3842:
3591:is also defined.
3437:
3415:
3381:
3366:
3359:
3344:
2928:Spherical vectors
2904:
2882:
2840:
2833:
2818:
2776:
2754:
2712:
2705:
2690:
2129:
2095:
2080:
1642:polar coordinates
1620:Polar coordinates
1570:
1545:
1520:
1424:
1371:
1318:
467:Arnold Sommerfeld
451:Hermann Grassmann
265:equivalence class
225:Giusto Bellavitis
192:
143:Euclidean vectors
78:of vector spaces.
59:Vector components
5263:
5228:
5227:
5110:Exterior algebra
5047:Hadamard product
4964:
4952:Linear equations
4823:
4816:
4809:
4800:
4799:
4794:
4793:
4773:
4767:
4766:
4764:
4763:
4748:
4742:
4732:
4721:
4710:
4704:
4695:
4689:
4676:
4670:
4651:
4645:
4630:Oliver Heaviside
4627:
4621:
4619:Internet Archive
4608:
4602:
4601:
4594:
4588:
4587:
4574:
4568:
4567:
4558:
4533:Euclidean vector
4523:
4521:
4520:
4515:
4484:
4482:
4481:
4476:
4445:
4443:
4442:
4437:
4410:
4408:
4407:
4402:
4400:
4398:
4387:
4385:
4377:
4375:
4364:
4362:
4354:
4352:
4341:
4339:
4302:
4300:
4299:
4296:{\displaystyle }
4294:
4289:
4281:
4256:
4254:
4253:
4248:
4246:
4238:
4220:
4218:
4217:
4212:
4210:
4202:
4182:
4180:
4179:
4174:
4172:
4171:
4166:
4135:
4133:
4132:
4127:
4125:
4117:
4093:
4091:
4090:
4085:
4083:
4082:
4077:
4057:
4055:
4054:
4049:
4044:
4036:
3997:
3995:
3994:
3989:
3987:
3982:
3977:
3963:
3961:
3960:
3955:
3950:
3919:
3917:
3916:
3911:
3909:
3902:
3888:
3886:
3885:
3880:
3878:
3873:
3868:
3858:
3856:
3855:
3850:
3848:
3843:
3835:
3805:
3803:
3802:
3797:
3795:
3787:
3774:
3772:
3771:
3766:
3764:
3753:
3715:
3713:
3712:
3707:
3705:
3686:
3684:
3683:
3678:
3676:
3639:
3637:
3636:
3631:
3629:
3621:
3586:
3584:
3583:
3578:
3576:
3575:
3570:
3557:
3555:
3554:
3549:
3547:
3546:
3541:
3523:
3521:
3520:
3515:
3513:
3512:
3507:
3463:
3461:
3460:
3455:
3453:
3452:
3435:
3431:
3430:
3413:
3392:
3390:
3389:
3384:
3382:
3374:
3364:
3360:
3352:
3342:
3273:
3271:
3270:
3265:
3263:
3262:
3221:
3208:
3206:
3205:
3200:
3198:
3197:
3160:
3147:
3145:
3144:
3139:
3107:
3094:
3092:
3091:
3086:
3054:
3032:
3030:
3029:
3024:
2923:
2921:
2920:
2915:
2902:
2898:
2897:
2880:
2859:
2857:
2856:
2851:
2838:
2834:
2826:
2816:
2795:
2793:
2792:
2787:
2774:
2770:
2769:
2752:
2731:
2729:
2728:
2723:
2710:
2706:
2698:
2688:
2606:
2604:
2603:
2598:
2547:
2545:
2544:
2539:
2537:
2536:
2498:
2485:
2483:
2482:
2477:
2475:
2474:
2440:
2427:
2425:
2424:
2419:
2390:
2377:
2375:
2374:
2369:
2340:
2318:
2316:
2315:
2310:
2274:
2272:
2271:
2266:
2155:
2153:
2152:
2147:
2145:
2144:
2127:
2106:
2104:
2103:
2098:
2096:
2088:
2078:
2032:
2030:
2029:
2024:
2001:
1999:
1998:
1993:
1991:
1990:
1959:
1946:
1944:
1943:
1938:
1936:
1935:
1906:
1893:
1891:
1890:
1885:
1862:
1849:
1847:
1846:
1841:
1818:
1796:
1794:
1793:
1788:
1745:
1743:
1742:
1737:
1735:
1734:
1706:
1704:
1703:
1698:
1582:
1580:
1579:
1574:
1572:
1571:
1563:
1560:
1559:
1547:
1546:
1538:
1535:
1534:
1522:
1521:
1513:
1510:
1509:
1497:
1485:
1483:
1482:
1477:
1475:
1460:
1458:
1457:
1452:
1426:
1425:
1417:
1407:
1405:
1404:
1399:
1373:
1372:
1364:
1354:
1352:
1351:
1346:
1320:
1319:
1311:
1298:
1296:
1295:
1290:
1288:
1287:
1282:
1256:
1254:
1253:
1248:
1246:
1245:
1240:
1227:
1225:
1224:
1219:
1217:
1216:
1211:
1146:
1144:
1143:
1138:
1136:
1135:
1128:
1127:
1114:
1113:
1087:
1086:
1073:
1072:
1052:
1051:
1044:
1043:
1030:
1029:
1003:
1002:
989:
988:
968:
955:
953:
952:
947:
945:
944:
937:
936:
925:
924:
902:
901:
890:
889:
869:
868:
861:
860:
849:
848:
826:
825:
814:
813:
793:
778:
776:
775:
770:
768:
737:
735:
734:
729:
727:
726:
721:
700:
698:
697:
692:
668:
666:
665:
660:
655:
654:
642:
641:
617:
616:
604:
603:
588:
442:. Subsequently,
413:Oliver Heaviside
262:
260:
259:
254:
204:
202:
201:
196:
194:
193:
185:
175:
165:
107:
89:
55:
32:
5271:
5270:
5266:
5265:
5264:
5262:
5261:
5260:
5241:
5240:
5239:
5234:
5216:
5178:
5134:
5071:
5023:
4965:
4956:
4922:Change of basis
4912:Multilinear map
4850:
4832:
4827:
4797:
4790:
4774:
4770:
4761:
4759:
4757:www.andrews.edu
4749:
4745:
4741:, third edition
4733:
4724:
4711:
4707:
4696:
4692:
4677:
4673:
4652:
4648:
4628:
4624:
4609:
4605:
4596:
4595:
4591:
4583:Vector Analysis
4575:
4571:
4560:
4559:
4555:
4551:
4529:
4500:
4497:
4496:
4461:
4458:
4457:
4424:
4421:
4420:
4391:
4386:
4381:
4368:
4363:
4358:
4345:
4340:
4335:
4333:
4330:
4329:
4318:
4310:Main articles:
4308:
4285:
4277:
4272:
4269:
4268:
4242:
4234:
4232:
4229:
4228:
4206:
4198:
4196:
4193:
4192:
4167:
4162:
4161:
4159:
4156:
4155:
4148:
4121:
4113:
4111:
4108:
4107:
4078:
4073:
4072:
4070:
4067:
4066:
4063:
4040:
4032:
4027:
4024:
4023:
4008:
3983:
3978:
3973:
3971:
3968:
3967:
3946:
3941:
3938:
3937:
3926:
3898:
3897:
3895:
3892:
3891:
3869:
3867:
3865:
3862:
3861:
3844:
3834:
3832:
3829:
3828:
3810:Scalar division
3791:
3783:
3781:
3778:
3777:
3760:
3749:
3747:
3744:
3743:
3722:
3701:
3693:
3690:
3689:
3672:
3667:
3664:
3663:
3645:
3625:
3617:
3615:
3612:
3611:
3600:Vector addition
3597:
3595:Vector addition
3571:
3566:
3565:
3563:
3560:
3559:
3542:
3537:
3536:
3534:
3531:
3530:
3508:
3503:
3502:
3500:
3497:
3496:
3476:
3470:
3448:
3444:
3426:
3422:
3399:
3396:
3395:
3373:
3351:
3328:
3325:
3324:
3295:
3293:Direct notation
3257:
3256:
3247:
3246:
3237:
3236:
3226:
3225:
3217:
3215:
3212:
3211:
3192:
3191:
3183:
3175:
3165:
3164:
3156:
3154:
3151:
3150:
3103:
3101:
3098:
3097:
3050:
3048:
3045:
3044:
3018:
3015:
3014:
3010:
2930:
2893:
2889:
2866:
2863:
2862:
2825:
2802:
2799:
2798:
2765:
2761:
2738:
2735:
2734:
2697:
2674:
2671:
2670:
2613:
2611:Direct notation
2607:) is optional.
2592:
2589:
2588:
2531:
2530:
2524:
2523:
2514:
2513:
2503:
2502:
2494:
2492:
2489:
2488:
2469:
2468:
2463:
2455:
2445:
2444:
2436:
2434:
2431:
2430:
2386:
2384:
2381:
2380:
2336:
2334:
2331:
2330:
2304:
2301:
2300:
2293:
2245:
2242:
2241:
2192: = 4.
2162:
2140:
2136:
2113:
2110:
2109:
2087:
2064:
2061:
2060:
2039:
2037:Direct notation
2033:) is optional.
2018:
2015:
2014:
1985:
1984:
1975:
1974:
1964:
1963:
1955:
1953:
1950:
1949:
1930:
1929:
1921:
1911:
1910:
1902:
1900:
1897:
1896:
1858:
1856:
1853:
1852:
1814:
1812:
1809:
1808:
1782:
1779:
1778:
1776:
1765:
1752:
1730:
1726:
1712:
1709:
1708:
1677:
1674:
1673:
1632:and polar axis
1622:
1611:
1602:
1593:
1562:
1561:
1555:
1551:
1537:
1536:
1530:
1526:
1512:
1511:
1505:
1501:
1493:
1491:
1488:
1487:
1471:
1469:
1466:
1465:
1416:
1415:
1413:
1410:
1409:
1363:
1362:
1360:
1357:
1356:
1310:
1309:
1307:
1304:
1303:
1283:
1278:
1277:
1275:
1272:
1271:
1265:
1241:
1236:
1235:
1233:
1230:
1229:
1212:
1207:
1206:
1204:
1201:
1200:
1197:
1182:
1173:
1163:
1156:
1130:
1129:
1123:
1119:
1116:
1115:
1103:
1099:
1096:
1095:
1089:
1088:
1082:
1078:
1075:
1074:
1068:
1064:
1057:
1056:
1046:
1045:
1039:
1035:
1032:
1031:
1019:
1015:
1012:
1011:
1005:
1004:
998:
994:
991:
990:
984:
980:
973:
972:
964:
962:
959:
958:
939:
938:
932:
928:
926:
914:
910:
908:
903:
897:
893:
891:
885:
881:
874:
873:
863:
862:
856:
852:
850:
838:
834:
832:
827:
821:
817:
815:
809:
805:
798:
797:
789:
787:
784:
783:
764:
762:
759:
758:
722:
717:
716:
714:
711:
710:
707:
705:Matrix notation
680:
677:
676:
650:
646:
631:
627:
612:
608:
599:
595:
584:
582:
579:
578:
556:
540:
518:
455:German language
405:Vector Analysis
400:vector products
388:Yale University
386:. Lecturing at
356:can be written
348:of two vectors
236:
233:
232:
221:
184:
183:
181:
178:
177:
171:
161:
141:, which may be
131:vector notation
119:
118:
117:
116:
115:
113:
108:
99:
98:
97:
95:
90:
81:
80:
79:
61:
56:
47:
46:
45:
36:
33:
24:
23:
22:Vector notation
17:
12:
11:
5:
5269:
5259:
5258:
5253:
5236:
5235:
5233:
5232:
5221:
5218:
5217:
5215:
5214:
5209:
5204:
5199:
5194:
5192:Floating-point
5188:
5186:
5180:
5179:
5177:
5176:
5174:Tensor product
5171:
5166:
5161:
5159:Function space
5156:
5151:
5145:
5143:
5136:
5135:
5133:
5132:
5127:
5122:
5117:
5112:
5107:
5102:
5097:
5095:Triple product
5092:
5087:
5081:
5079:
5073:
5072:
5070:
5069:
5064:
5059:
5054:
5049:
5044:
5039:
5033:
5031:
5025:
5024:
5022:
5021:
5016:
5011:
5009:Transformation
5006:
5001:
4999:Multiplication
4996:
4991:
4986:
4981:
4975:
4973:
4967:
4966:
4959:
4957:
4955:
4954:
4949:
4944:
4939:
4934:
4929:
4924:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4864:
4858:
4856:
4855:Basic concepts
4852:
4851:
4849:
4848:
4843:
4837:
4834:
4833:
4830:Linear algebra
4826:
4825:
4818:
4811:
4803:
4796:
4795:
4788:
4768:
4743:
4722:
4705:
4690:
4671:
4646:
4622:
4603:
4589:
4569:
4552:
4550:
4547:
4546:
4545:
4540:
4535:
4528:
4525:
4513:
4510:
4507:
4504:
4495:is written as
4474:
4471:
4468:
4465:
4456:is written as
4435:
4431:
4428:
4419:is written as
4397:
4394:
4390:
4384:
4380:
4374:
4371:
4367:
4361:
4357:
4351:
4348:
4344:
4338:
4326:Nabla operator
4307:
4304:
4292:
4288:
4284:
4280:
4276:
4245:
4241:
4237:
4209:
4205:
4201:
4170:
4165:
4147:
4144:
4140:dyadic product
4124:
4120:
4116:
4081:
4076:
4062:
4059:
4047:
4043:
4039:
4035:
4031:
4007:
4004:
4000:absolute value
3986:
3981:
3976:
3953:
3949:
3945:
3925:
3922:
3921:
3920:
3908:
3905:
3901:
3889:
3876:
3872:
3859:
3847:
3841:
3838:
3807:
3806:
3794:
3790:
3786:
3775:
3763:
3759:
3756:
3752:
3721:
3718:
3717:
3716:
3704:
3700:
3697:
3687:
3675:
3671:
3655:with a vector
3644:
3641:
3628:
3624:
3620:
3596:
3593:
3574:
3569:
3545:
3540:
3511:
3506:
3469:
3466:
3465:
3464:
3451:
3447:
3443:
3440:
3434:
3429:
3425:
3421:
3418:
3412:
3409:
3406:
3403:
3393:
3380:
3377:
3372:
3369:
3363:
3358:
3355:
3350:
3347:
3341:
3338:
3335:
3332:
3294:
3291:
3275:
3274:
3261:
3255:
3252:
3249:
3248:
3245:
3242:
3239:
3238:
3235:
3232:
3231:
3229:
3224:
3220:
3209:
3196:
3190:
3187:
3184:
3182:
3179:
3176:
3174:
3171:
3170:
3168:
3163:
3159:
3148:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3106:
3095:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3053:
3022:
3009:
3006:
2929:
2926:
2925:
2924:
2913:
2910:
2907:
2901:
2896:
2892:
2888:
2885:
2879:
2876:
2873:
2870:
2860:
2849:
2846:
2843:
2837:
2832:
2829:
2824:
2821:
2815:
2812:
2809:
2806:
2796:
2785:
2782:
2779:
2773:
2768:
2764:
2760:
2757:
2751:
2748:
2745:
2742:
2732:
2721:
2718:
2715:
2709:
2704:
2701:
2696:
2693:
2687:
2684:
2681:
2678:
2612:
2609:
2596:
2549:
2548:
2535:
2529:
2526:
2525:
2522:
2519:
2516:
2515:
2512:
2509:
2508:
2506:
2501:
2497:
2486:
2473:
2467:
2464:
2462:
2459:
2456:
2454:
2451:
2450:
2448:
2443:
2439:
2428:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2389:
2378:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2339:
2308:
2292:
2289:
2264:
2261:
2258:
2255:
2252:
2249:
2161:
2158:
2157:
2156:
2143:
2139:
2135:
2132:
2126:
2123:
2120:
2117:
2107:
2094:
2091:
2086:
2083:
2077:
2074:
2071:
2068:
2038:
2035:
2022:
2003:
2002:
1989:
1983:
1980:
1977:
1976:
1973:
1970:
1969:
1967:
1962:
1958:
1947:
1934:
1928:
1925:
1922:
1920:
1917:
1916:
1914:
1909:
1905:
1894:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1861:
1850:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1817:
1786:
1774:
1763:
1751:
1748:
1733:
1729:
1725:
1722:
1719:
1716:
1696:
1693:
1690:
1687:
1684:
1681:
1621:
1618:
1607:
1598:
1589:
1569:
1566:
1558:
1554:
1550:
1544:
1541:
1533:
1529:
1525:
1519:
1516:
1508:
1504:
1500:
1496:
1474:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1423:
1420:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1370:
1367:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1317:
1314:
1286:
1281:
1261:
1244:
1239:
1215:
1210:
1196:
1193:
1178:
1172: − 1
1168:
1161:
1154:
1148:
1147:
1134:
1126:
1122:
1118:
1117:
1112:
1109:
1106:
1102:
1098:
1097:
1094:
1091:
1090:
1085:
1081:
1077:
1076:
1071:
1067:
1063:
1062:
1060:
1055:
1050:
1042:
1038:
1034:
1033:
1028:
1025:
1022:
1018:
1014:
1013:
1010:
1007:
1006:
1001:
997:
993:
992:
987:
983:
979:
978:
976:
971:
967:
956:
943:
935:
931:
927:
923:
920:
917:
913:
909:
907:
904:
900:
896:
892:
888:
884:
880:
879:
877:
872:
867:
859:
855:
851:
847:
844:
841:
837:
833:
831:
828:
824:
820:
816:
812:
808:
804:
803:
801:
796:
792:
767:
725:
720:
706:
703:
690:
687:
684:
674:angle brackets
670:
669:
658:
653:
649:
645:
640:
637:
634:
630:
626:
623:
620:
615:
611:
607:
602:
598:
594:
591:
587:
555:
554:Tuple notation
552:
517:
514:
471:Academic Press
465:, he assigned
425:Gothic letters
396:scalar product
378:W. K. Clifford
276:W. R. Hamilton
274:was coined by
252:
249:
246:
243:
240:
220:
217:
191:
188:
111:Vector product
109:
102:
101:
100:
93:Scalar product
91:
84:
83:
82:
57:
50:
49:
48:
37:Pointing from
34:
27:
26:
25:
21:
20:
19:
18:
15:
9:
6:
4:
3:
2:
5268:
5257:
5254:
5252:
5249:
5248:
5246:
5231:
5223:
5222:
5219:
5213:
5210:
5208:
5207:Sparse matrix
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5189:
5187:
5185:
5181:
5175:
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5146:
5144:
5142:constructions
5141:
5137:
5131:
5130:Outermorphism
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5090:Cross product
5088:
5086:
5083:
5082:
5080:
5078:
5074:
5068:
5065:
5063:
5060:
5058:
5057:Outer product
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5037:Orthogonality
5035:
5034:
5032:
5030:
5026:
5020:
5017:
5015:
5014:Cramer's rule
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4995:
4992:
4990:
4987:
4985:
4984:Decomposition
4982:
4980:
4977:
4976:
4974:
4972:
4968:
4963:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4925:
4923:
4920:
4918:
4915:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4859:
4857:
4853:
4847:
4844:
4842:
4839:
4838:
4835:
4831:
4824:
4819:
4817:
4812:
4810:
4805:
4804:
4801:
4791:
4789:9780486161167
4785:
4781:
4780:
4772:
4758:
4754:
4747:
4740:
4736:
4731:
4729:
4727:
4720:
4716:
4715:
4709:
4703:
4699:
4694:
4688:
4684:
4680:
4675:
4668:
4664:
4660:
4656:
4650:
4643:
4639:
4635:
4631:
4626:
4620:
4616:
4612:
4607:
4599:
4593:
4585:
4584:
4579:
4573:
4565:
4564:
4557:
4553:
4544:
4541:
4539:
4536:
4534:
4531:
4530:
4524:
4511:
4508:
4505:
4494:
4490:
4485:
4472:
4469:
4466:
4455:
4451:
4446:
4433:
4429:
4418:
4414:
4395:
4378:
4372:
4355:
4349:
4327:
4323:
4317:
4313:
4303:
4282:
4266:
4262:
4257:
4239:
4226:
4225:wedge product
4221:
4203:
4190:
4186:
4168:
4153:
4152:cross product
4146:Cross product
4143:
4141:
4136:
4118:
4105:
4101:
4097:
4079:
4058:
4037:
4021:
4017:
4013:
4012:inner product
4006:Inner product
4003:
4001:
3964:
3935:
3931:
3906:
3903:
3890:
3874:
3860:
3839:
3836:
3827:
3826:
3825:
3823:
3820:and a scalar
3819:
3815:
3811:
3788:
3776:
3757:
3754:
3742:
3741:
3740:
3738:
3734:
3729:
3725:
3698:
3695:
3688:
3669:
3662:
3661:
3660:
3658:
3654:
3649:
3640:
3622:
3609:
3605:
3601:
3592:
3590:
3589:cross product
3572:
3543:
3528:
3524:
3509:
3493:
3489:
3485:
3481:
3478:In any given
3475:
3449:
3445:
3441:
3438:
3432:
3427:
3423:
3419:
3416:
3410:
3407:
3404:
3401:
3394:
3378:
3375:
3370:
3367:
3361:
3356:
3353:
3348:
3345:
3339:
3336:
3333:
3330:
3323:
3322:
3321:
3319:
3315:
3310:
3308:
3304:
3300:
3290:
3288:
3284:
3280:
3259:
3253:
3243:
3233:
3227:
3222:
3210:
3194:
3188:
3180:
3172:
3166:
3161:
3149:
3132:
3126:
3123:
3117:
3114:
3108:
3096:
3079:
3073:
3070:
3064:
3061:
3055:
3043:
3042:
3041:
3039:
3034:
3005:
3004:(exclusive).
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2971:
2967:
2963:
2959:
2955:
2951:
2947:
2943:
2939:
2934:
2911:
2908:
2905:
2899:
2894:
2890:
2886:
2883:
2877:
2874:
2871:
2868:
2861:
2847:
2844:
2841:
2835:
2830:
2827:
2822:
2819:
2813:
2810:
2807:
2804:
2797:
2783:
2780:
2777:
2771:
2766:
2762:
2758:
2755:
2749:
2746:
2743:
2740:
2733:
2719:
2716:
2713:
2707:
2702:
2699:
2694:
2691:
2685:
2682:
2679:
2676:
2669:
2668:
2667:
2665:
2661:
2657:
2653:
2648:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2608:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2533:
2527:
2520:
2510:
2504:
2499:
2487:
2471:
2465:
2460:
2452:
2446:
2441:
2429:
2412:
2409:
2406:
2400:
2397:
2391:
2379:
2362:
2359:
2356:
2350:
2347:
2341:
2329:
2328:
2327:
2325:
2320:
2298:
2288:
2286:
2282:
2278:
2262:
2259:
2256:
2253:
2250:
2247:
2239:
2235:
2231:
2227:
2223:
2219:
2215:
2211:
2207:
2203:
2199:
2191:
2187:
2183:
2179:
2175:
2172:, polar axis
2171:
2166:
2141:
2137:
2133:
2130:
2124:
2121:
2118:
2115:
2108:
2092:
2089:
2084:
2081:
2075:
2072:
2069:
2066:
2059:
2058:
2057:
2055:
2050:
2048:
2044:
2034:
2012:
2008:
1987:
1981:
1971:
1965:
1960:
1948:
1932:
1926:
1918:
1912:
1907:
1895:
1878:
1872:
1869:
1863:
1851:
1834:
1828:
1825:
1819:
1807:
1806:
1805:
1803:
1798:
1773:
1769:
1762:
1758:
1747:
1731:
1727:
1723:
1720:
1717:
1714:
1694:
1691:
1688:
1685:
1682:
1679:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1635:
1631:
1626:
1617:
1615:
1610:
1606:
1601:
1597:
1592:
1588:
1583:
1556:
1552:
1548:
1531:
1527:
1523:
1506:
1502:
1498:
1462:
1445:
1442:
1439:
1436:
1433:
1427:
1392:
1389:
1386:
1383:
1380:
1374:
1339:
1336:
1333:
1330:
1327:
1321:
1302:
1284:
1269:
1264:
1260:
1242:
1213:
1192:
1190:
1186:
1181:
1177:
1171:
1167:
1160:
1153:
1132:
1124:
1120:
1110:
1107:
1104:
1100:
1092:
1083:
1079:
1069:
1065:
1058:
1053:
1048:
1040:
1036:
1026:
1023:
1020:
1016:
1008:
999:
995:
985:
981:
974:
969:
957:
941:
933:
929:
921:
918:
915:
911:
905:
898:
894:
886:
882:
875:
870:
865:
857:
853:
845:
842:
839:
835:
829:
822:
818:
810:
806:
799:
794:
782:
781:
780:
756:
751:
749:
748:column vector
745:
741:
723:
702:
685:
675:
651:
647:
643:
638:
635:
632:
628:
624:
621:
618:
613:
609:
605:
600:
596:
589:
577:
576:
575:
573:
569:
566:-dimensional
565:
561:
551:
549:
545:
539:
530:
522:
512:
507:
505:
499:
494:
492:
488:
484:
482:
478:
477:
472:
468:
464:
460:
456:
452:
447:
445:
441:
437:
432:
430:
426:
422:
421:Greek letters
418:
414:
409:
407:
406:
401:
397:
393:
389:
385:
384:
379:
375:
371:
367:
363:
359:
355:
351:
347:
343:
339:
338:cross product
335:
331:
327:
323:
320:
316:
312:
308:
305:
301:
297:
293:
289:
285:
281:
277:
273:
268:
266:
250:
247:
244:
241:
238:
230:
226:
216:
214:
210:
208:
186:
174:
169:
164:
159:
154:
152:
148:
144:
140:
136:
132:
128:
124:
112:
106:
94:
88:
77:
73:
69:
65:
60:
54:
44:
40:
31:
5140:Vector space
4872:Vector space
4778:
4771:
4760:. Retrieved
4756:
4746:
4738:
4719:Google Books
4717:, p. 10, at
4712:
4708:
4693:
4686:
4674:
4658:
4649:
4633:
4625:
4606:
4592:
4582:
4572:
4562:
4556:
4488:
4486:
4449:
4447:
4412:
4325:
4324:through the
4319:
4316:Nabla symbol
4264:
4260:
4258:
4222:
4188:
4184:
4149:
4137:
4103:
4099:
4064:
4019:
4015:
4009:
3965:
3933:
3927:
3821:
3817:
3809:
3808:
3736:
3732:
3727:
3726:
3723:
3656:
3652:
3646:
3607:
3603:
3598:
3480:vector space
3477:
3317:
3313:
3311:
3306:
3302:
3298:
3296:
3286:
3282:
3278:
3276:
3037:
3035:
3011:
3001:
2997:
2993:
2989:
2985:
2981:
2975:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2663:
2659:
2655:
2651:
2649:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2616:
2614:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2550:
2323:
2321:
2297:concatenated
2294:
2284:
2280:
2276:
2237:
2229:
2225:
2221:
2213:
2209:
2205:
2201:
2195:
2189:
2185:
2181:
2177:
2173:
2169:
2053:
2051:
2046:
2042:
2040:
2010:
2006:
2004:
1801:
1799:
1771:
1770:(instead of
1767:
1760:
1759:(instead of
1756:
1753:
1669:
1657:
1649:
1639:
1633:
1629:
1613:
1608:
1604:
1599:
1595:
1590:
1586:
1584:
1463:
1301:unit vectors
1262:
1258:
1199:A vector in
1198:
1184:
1179:
1175:
1169:
1165:
1158:
1151:
1149:
754:
752:
709:A vector in
708:
671:
563:
559:
557:
541:
509:
501:
496:
490:
485:
480:
474:
448:
435:
433:
423:by Tait and
410:
403:
381:
373:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
321:
318:
314:
310:
306:
303:
299:
295:
291:
287:
283:
271:
269:
222:
212:
211:
172:
162:
155:
151:vector space
130:
120:
71:
67:
63:
42:
38:
35:Vector arrow
5120:Multivector
5085:Determinant
5042:Dot product
4887:Linear span
4735:Felix Klein
4698:Karin Reich
4667:Hathi Trust
4096:dot product
4061:Dot product
3527:dot product
2950:mathematics
2647:and so on.
1707:radians or
548:coordinates
487:Felix Klein
459:Felix Klein
415:argued for
376:. In 1878,
342:dot product
280:quaternions
229:equipollent
123:mathematics
76:isomorphism
5245:Categories
5154:Direct sum
4989:Invertible
4892:Linear map
4762:2023-07-25
4549:References
4454:divergence
3468:Operations
2571:-axis and
1191:for more.
753:Again, an
744:row vector
672:Sometimes
536:See also:
74:yields an
5184:Numerical
4947:Transpose
4506:×
4503:∇
4467:⋅
4464:∇
4427:∇
4393:∂
4389:∂
4370:∂
4366:∂
4347:∂
4343:∂
4240:∧
4204:×
4119:⋅
4046:⟩
4030:⟨
3952:‖
3944:‖
3904:÷
3789:−
3758:−
3699:⋅
3450:∘
3439:ϕ
3428:∘
3417:θ
3402:ρ
3376:π
3368:ϕ
3354:π
3346:θ
3331:ρ
3254:ϕ
3251:∠
3244:θ
3241:∠
3234:ρ
3189:ϕ
3186:∠
3181:θ
3178:∠
3173:ρ
3136:⟩
3133:ϕ
3130:∠
3124:θ
3121:∠
3115:ρ
3112:⟨
3080:ϕ
3077:∠
3071:θ
3068:∠
3062:ρ
3021:∠
2895:∘
2884:ϕ
2869:ρ
2828:π
2820:ϕ
2805:ρ
2767:∘
2756:θ
2700:π
2692:θ
2658:-axis is
2595:∠
2559:onto the
2521:θ
2518:∠
2461:θ
2458:∠
2416:⟩
2407:θ
2404:∠
2395:⟨
2357:θ
2354:∠
2307:∠
2263:π
2254:θ
2251:≤
2142:∘
2131:θ
2090:π
2082:θ
2021:∠
1982:θ
1979:∠
1927:θ
1924:∠
1882:⟩
1879:θ
1876:∠
1867:⟨
1835:θ
1832:∠
1785:∠
1732:∘
1721:θ
1718:≤
1695:π
1686:θ
1683:≤
1646:magnitude
1568:^
1543:^
1540:ȷ
1518:^
1515:ı
1422:^
1369:^
1366:ȷ
1316:^
1313:ı
1108:−
1093:⋮
1024:−
1009:⋮
919:−
906:⋯
843:−
830:⋯
689:⟩
686:…
683:⟨
636:−
622:…
558:A vector
524:Rectangle
417:Clarendon
411:In 1891,
344:(.), the
270:The term
245:≏
190:→
5230:Category
5169:Subspace
5164:Quotient
5115:Bivector
5029:Bilinear
4971:Matrices
4846:Glossary
4659:Bulletin
4580:(1911).
4527:See also
4417:gradient
4322:calculus
2563:-plane,
1640:The two
542:Given a
436:Bulletin
340:(×) and
223:In 1835
207:variable
135:notation
4841:Outline
4700:(1995)
4681:(1912)
4661:of the
4613:(1901)
2631:), and
1664:letter
438:of the
429:Maxwell
219:History
147:members
139:vectors
127:physics
5125:Tensor
4937:Kernel
4867:Vector
4862:Scalar
4786:
4543:Phasor
4491:, the
4452:, the
4415:, the
3436:
3414:
3365:
3343:
3305:, and
3277:Where
2903:
2881:
2839:
2817:
2775:
2753:
2711:
2689:
2575:, and
2551:Where
2128:
2079:
2005:where
1654:origin
1603:, and
1585:where
1408:, and
1257:where
1150:where
740:matrix
317:, and
272:vector
166:. The
4994:Minor
4979:Block
4917:Basis
4685:from
4306:Nabla
3529:. In
1666:theta
1662:Greek
1660:(the
1268:basis
1164:, …,
572:tuple
149:of a
5149:Dual
5004:Rank
4784:ISBN
4665:via
4493:curl
4314:and
4263:and
4187:and
4150:The
4102:and
4018:and
4010:The
3930:norm
3928:The
3924:Norm
3735:and
3606:and
3558:and
3488:norm
2968:and
2635:(or
2627:(or
2619:(or
2257:<
2045:and
1724:<
1689:<
398:and
352:and
332:j +
328:i +
298:j +
294:i +
125:and
70:and
4642:alt
4638:109
4617:at
4312:Del
4065:In
2623:),
2279:or
2234:phi
2228:or
2218:rho
2212:or
1728:360
1270:in
562:in
481:and
427:by
360:= –
121:In
41:to
5247::
4755:.
4725:^
4657:,
4632:,
4328::
4267::
3446:45
3424:20
3309:.
3301:,
2944:,
2940:,
2891:20
2763:20
2664:xy
2652:xy
2581:xy
2561:xy
2285:xy
2222:xy
2206:xy
2202:xy
2138:20
1797:.
1746:.
1594:,
1461:.
1355:,
1174:,
1157:,
750:.
550:.
431:.
408:.
390:,
368:+
358:pq
324:=
309:=
290:+
286:=
209:.
153:.
129:,
4822:e
4815:t
4808:v
4792:.
4765:.
4669:.
4644:)
4640:(
4512:.
4509:F
4489:F
4473:,
4470:F
4450:F
4434:,
4430:f
4413:f
4396:z
4383:k
4379:+
4373:y
4360:j
4356:+
4350:x
4337:i
4291:]
4287:v
4283:,
4279:u
4275:[
4265:v
4261:u
4244:v
4236:u
4208:v
4200:u
4189:v
4185:u
4169:3
4164:R
4123:v
4115:u
4104:v
4100:u
4080:n
4075:R
4042:v
4038:,
4034:u
4020:v
4016:u
3985:|
3980:v
3975:|
3948:v
3934:v
3907:c
3900:v
3875:c
3871:v
3846:v
3840:c
3837:1
3822:c
3818:v
3793:v
3785:u
3762:v
3755:+
3751:u
3737:v
3733:u
3703:v
3696:k
3674:v
3670:k
3657:v
3653:k
3627:v
3623:+
3619:u
3608:v
3604:u
3573:7
3568:R
3544:3
3539:R
3510:n
3505:R
3442:=
3433:,
3420:=
3411:,
3408:5
3405:=
3379:4
3371:=
3362:,
3357:9
3349:=
3340:,
3337:5
3334:=
3318:π
3314:π
3307:φ
3303:θ
3299:ρ
3287:φ
3283:θ
3279:ρ
3260:]
3228:[
3223:=
3219:v
3195:]
3167:[
3162:=
3158:v
3127:,
3118:,
3109:=
3105:v
3083:)
3074:,
3065:,
3059:(
3056:=
3052:v
3038:v
3013:(
3002:π
2998:z
2994:φ
2990:x
2986:θ
2982:ρ
2970:φ
2966:θ
2962:φ
2958:θ
2954:r
2946:φ
2942:θ
2938:r
2912:3
2909:=
2906:z
2900:,
2887:=
2878:,
2875:5
2872:=
2848:3
2845:=
2842:z
2836:,
2831:9
2823:=
2814:,
2811:5
2808:=
2784:3
2781:=
2778:h
2772:,
2759:=
2750:,
2747:5
2744:=
2741:r
2720:3
2717:=
2714:h
2708:,
2703:9
2695:=
2686:,
2683:5
2680:=
2677:r
2660:π
2656:x
2645:θ
2641:ρ
2637:z
2633:h
2629:φ
2625:θ
2621:ρ
2617:r
2585:v
2577:h
2573:v
2569:x
2565:θ
2557:v
2553:r
2534:]
2528:h
2511:r
2505:[
2500:=
2496:v
2472:]
2466:h
2453:r
2447:[
2442:=
2438:v
2413:h
2410:,
2401:,
2398:r
2392:=
2388:v
2366:)
2363:h
2360:,
2351:,
2348:r
2345:(
2342:=
2338:v
2324:v
2281:z
2277:h
2260:2
2248:0
2238:x
2230:φ
2226:θ
2214:ρ
2210:r
2190:z
2186:φ
2182:ρ
2178:L
2174:A
2170:O
2134:=
2125:,
2122:5
2119:=
2116:r
2093:9
2085:=
2076:,
2073:5
2070:=
2067:r
2054:π
2047:θ
2043:r
2011:θ
2007:r
1988:]
1972:r
1966:[
1961:=
1957:v
1933:]
1919:r
1913:[
1908:=
1904:v
1873:,
1870:r
1864:=
1860:v
1838:)
1829:,
1826:r
1823:(
1820:=
1816:v
1802:v
1775:2
1772:v
1768:θ
1764:1
1761:v
1757:r
1715:0
1692:2
1680:0
1670:x
1658:θ
1650:r
1634:L
1630:O
1614:v
1609:z
1605:v
1600:y
1596:v
1591:x
1587:v
1565:k
1557:z
1553:v
1549:+
1532:y
1528:v
1524:+
1507:x
1503:v
1499:=
1495:v
1473:v
1449:)
1446:1
1443:,
1440:0
1437:,
1434:0
1431:(
1428:=
1419:k
1396:)
1393:0
1390:,
1387:1
1384:,
1381:0
1378:(
1375:=
1343:)
1340:0
1337:,
1334:0
1331:,
1328:1
1325:(
1322:=
1285:3
1280:R
1263:z
1259:v
1243:2
1238:R
1214:3
1209:R
1185:v
1180:n
1176:v
1170:n
1166:v
1162:2
1159:v
1155:1
1152:v
1133:)
1125:n
1121:v
1111:1
1105:n
1101:v
1084:2
1080:v
1070:1
1066:v
1059:(
1054:=
1049:]
1041:n
1037:v
1027:1
1021:n
1017:v
1000:2
996:v
986:1
982:v
975:[
970:=
966:v
942:)
934:n
930:v
922:1
916:n
912:v
899:2
895:v
887:1
883:v
876:(
871:=
866:]
858:n
854:v
846:1
840:n
836:v
823:2
819:v
811:1
807:v
800:[
795:=
791:v
766:v
755:n
724:n
719:R
657:)
652:n
648:v
644:,
639:1
633:n
629:v
625:,
619:,
614:2
610:v
606:,
601:1
597:v
593:(
590:=
586:v
564:n
560:v
374:q
372:×
370:p
366:q
364:.
362:p
354:q
350:p
334:d
330:c
326:b
322:q
319:V
315:q
311:a
307:q
304:S
300:d
296:c
292:b
288:a
284:q
251:D
248:C
242:B
239:A
187:v
173:v
163:v
72:y
68:x
64:v
43:B
39:A
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