147:
22:
1441:
119:
1279:
1798:
1436:{\displaystyle s=\left\|\mathbf {a} _{1}\right\|=\left\|\mathbf {a} \right\|\cos \theta =\left\|\mathbf {a} \right\|{\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}\,}
1248:
372:
982:
1760:
1664:
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1025:
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1169:
488:
430:
761:
1532:
1136:
517:
459:
298:
243:
1111:
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909:
887:
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732:
710:
685:
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218:
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395:
1271:
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1716:
58:
1837:
1620:
1454:
1243:{\displaystyle {\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|}}=\cos \theta }
1803:
65:
1670:
1577:
987:
39:
1537:
105:
1144:
72:
463:
1859:
1793:
43:
54:
404:
737:
1508:
1116:
497:
439:
278:
223:
1094:
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870:
848:
792:
770:
715:
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668:
646:
624:
568:
546:
252:
201:
32:
367:{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,}
599:
79:
1074:
1050:
828:
618:
595:
522:
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610:
272:
8:
1256:
196:
1833:
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764:
150:
139:
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1505:
if the angle is smaller than 90°. More exactly, if the vector projection is denoted
665:, with a negative sign if the projection has an opposite direction with respect to
603:
1771:
1498:
614:
491:
135:
1853:
1776:
146:
1139:
1032:
433:
398:
188:
763:
converts it into the above-mentioned orthogonal projection, also called
21:
1068:
118:
823:
540:
1253:
By this property, the definition of the scalar projection
977:{\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta .}
1755:{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }.}
1659:{\displaystyle 0^{\circ }\leq \theta \leq 90^{\circ },}
1490:{\displaystyle 90^{\circ }<\theta \leq 180^{\circ }}
1804:
Scalar
Projection & Vector Projection - medium.com
1719:
1673:
1623:
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1511:
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1119:
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1053:
990:
942:
917:
895:
873:
851:
831:
795:
773:
740:
718:
696:
671:
649:
627:
571:
549:
525:
500:
466:
442:
407:
383:
309:
281:
255:
226:
204:
602:
are the scalar projections in the directions of the
46:. Unsourced material may be challenged and removed.
1754:
1706:{\displaystyle s=-\left\|\mathbf {a} _{1}\right\|}
1705:
1658:
1609:
1563:
1526:
1489:
1435:
1265:
1242:
1163:
1130:
1105:
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1059:
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976:
925:
903:
881:
859:
837:
803:
781:
755:
726:
704:
679:
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635:
579:
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531:
511:
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453:
424:
389:
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292:
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1610:{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
1042:
1020:{\displaystyle s=\left\|\mathbf {a} _{1}\right\|}
747:
355:
126:≤ 90°, as in this case, the scalar projection of
1851:
1031:The formula above can be inverted to obtain the
1564:{\displaystyle \left\|\mathbf {a} _{1}\right\|}
1164:{\displaystyle \mathbf {a} \cdot \mathbf {b} }
814:
594:refers sometimes to scalar projection, as, in
1451:The scalar projection has a negative sign if
1832:(5th ed.). Wellesley: Cambridge press.
1809:Lesson Explainer: Scalar Projection | Nagwa
483:{\displaystyle \left\|\mathbf {a} \right\|}
1432:
106:Learn how and when to remove this message
145:
117:
1852:
1827:
1799:Scalar projection - Flexbooks.ck12.org
690:Multiplying the scalar projection of
425:{\displaystyle {\hat {\mathbf {b} }}}
44:adding citations to reliable sources
15:
889:is known, the scalar projection of
756:{\displaystyle \mathbf {\hat {b}} }
13:
14:
1871:
1138:by the following property of the
1689:
1593:
1547:
1527:{\displaystyle \mathbf {a} _{1}}
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1412:
1404:
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1373:
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1204:
1193:
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1157:
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1099:
1003:
954:
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651:
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573:
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342:
321:
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257:
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20:
31:needs additional citations for
1830:Introduction to linear algebra
1821:
1699:
1684:
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1557:
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1426:
1418:
1390:
1382:
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1369:
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1043:Definition in terms of a and b
1013:
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1:
1814:
1446:
1131:{\displaystyle \mathbf {b} ,}
512:{\displaystyle \mathbf {a} ,}
454:{\displaystyle \mathbf {b} ,}
293:{\displaystyle \mathbf {b} ,}
238:{\displaystyle \mathbf {b} ,}
1106:{\displaystyle \mathbf {a} }
1091:can be computed in terms of
926:{\displaystyle \mathbf {b} }
904:{\displaystyle \mathbf {a} }
882:{\displaystyle \mathbf {b} }
860:{\displaystyle \mathbf {a} }
804:{\displaystyle \mathbf {b} }
782:{\displaystyle \mathbf {a} }
727:{\displaystyle \mathbf {b} }
705:{\displaystyle \mathbf {a} }
680:{\displaystyle \mathbf {b} }
658:{\displaystyle \mathbf {b} }
636:{\displaystyle \mathbf {a} }
580:{\displaystyle \mathbf {b} }
558:{\displaystyle \mathbf {a} }
264:{\displaystyle \mathbf {a} }
213:{\displaystyle \mathbf {a} }
7:
1765:
609:The scalar projection is a
168:), and vector rejection of
10:
1876:
1794:Dot products - www.mit.org
1787:
815:Definition based on angle
1828:Strang, Gilbert (2016).
1497:. It coincides with the
1084:{\displaystyle \theta }
1060:{\displaystyle \theta }
838:{\displaystyle \theta }
532:{\displaystyle \theta }
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933:can be computed using
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600:components of a vector
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390:{\displaystyle \cdot }
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220:on (or onto) a vector
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1860:Operations on vectors
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1501:of the corresponding
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619:orthogonal projection
596:Cartesian coordinates
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436:in the direction of
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40:improve this article
377:where the operator
134:coincides with the
55:"Scalar projection"
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245:also known as the
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1839:978-0-9802327-7-6
1782:Vector projection
1503:vector projection
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1266:{\displaystyle s}
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765:vector projection
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193:scalar projection
151:Vector projection
140:vector projection
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1534:and its length
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604:coordinate axes
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247:scalar resolute
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1772:Scalar product
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1027:in the figure)
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1777:Cross product
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300:is given by:
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88:
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78:
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67:
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60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
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1823:
1450:
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93:
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76:
69:
62:
50:
38:Please help
33:verification
30:
1140:dot product
434:unit vector
399:dot product
189:mathematics
1815:References
1447:Properties
397:denotes a
96:March 2024
66:newspapers
1745:∘
1737:≤
1734:θ
1726:∘
1681:−
1649:∘
1641:≤
1638:θ
1635:≤
1630:∘
1483:∘
1475:≤
1472:θ
1464:∘
1409:⋅
1359:⋅
1332:θ
1329:
1273:becomes:
1238:θ
1235:
1190:⋅
1154:⋅
1079:θ
1055:θ
969:θ
966:
833:θ
748:^
590:The term
527:θ
417:^
385:⋅
356:^
347:⋅
336:θ
333:
273:direction
1854:Category
1766:See also
1700:‖
1685:‖
1604:‖
1589:‖
1558:‖
1543:‖
1427:‖
1419:‖
1391:‖
1383:‖
1378:‖
1370:‖
1347:‖
1339:‖
1322:‖
1314:‖
1306:‖
1291:‖
1222:‖
1214:‖
1209:‖
1201:‖
1014:‖
999:‖
959:‖
951:‖
845:between
543:between
477:‖
469:‖
326:‖
318:‖
122:If 0° ≤
1788:Sources
822:If the
617:of the
539:is the
490:is the
432:is the
271:in the
138:of the
80:scholar
1836:
1499:length
1069:cosine
615:length
611:scalar
598:, the
492:length
197:vector
191:, the
136:length
82:
75:
68:
61:
53:
1047:When
1033:angle
824:angle
541:angle
195:of a
172:from
87:JSTOR
73:books
1834:ISBN
1731:<
1469:<
1113:and
867:and
565:and
519:and
59:news
1741:180
1713:if
1617:if
1479:180
1326:cos
1232:cos
1071:of
963:cos
911:on
789:on
767:of
734:by
712:on
643:on
621:of
494:of
330:cos
275:of
249:of
187:In
157:on
153:of
130:on
42:by
1856::
1722:90
1645:90
1571::
1460:90
1171::
1039:.
1035:,
811:.
687:.
606:.
587:.
401:,
183:).
1842:.
1750:.
1695:1
1690:a
1678:=
1675:s
1654:,
1626:0
1599:1
1594:a
1585:=
1582:s
1553:1
1548:a
1520:1
1515:a
1423:b
1413:b
1405:a
1398:=
1387:b
1374:a
1363:b
1355:a
1343:a
1335:=
1318:a
1310:=
1301:1
1296:a
1287:=
1284:s
1261:s
1229:=
1218:b
1205:a
1194:b
1186:a
1158:b
1150:a
1126:,
1122:b
1100:a
1037:θ
1009:1
1004:a
995:=
992:s
984:(
972:.
955:a
947:=
944:s
920:b
898:a
876:b
854:a
817:θ
798:b
776:a
745:b
721:b
699:a
674:b
652:b
630:a
574:b
552:a
507:,
503:a
473:a
449:,
445:b
413:b
362:,
353:b
343:a
339:=
322:a
314:=
311:s
288:,
284:b
258:a
233:,
229:b
207:a
181:2
178:a
176:(
174:b
170:a
166:1
163:a
161:(
159:b
155:a
142:.
132:b
128:a
124:θ
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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