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39:
quantity. The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second
282:
122:. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. So, if
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Vector multiplication has multiple applications in regards to mathematics, but also in other studies such as physics and engineering.
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277:{\displaystyle \mathbf {a} \times \mathbf {b} =|\mathbf {a} |\,|\mathbf {b} |\sin \theta \,\mathbf {\hat {n}} .}
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is a mixed quantity consisting of a scalar plus a bivector. The geometric product is well defined for any
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626:{\displaystyle \mathbf {a} \mathbf {b} =\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \wedge \mathbf {b} }
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36:
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incorrectly led you here, you may wish to change the link to point directly to the intended article.
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118:– also known as the "vector product", a binary operation on two vectors that results in another
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35:– also known as the "scalar product", a binary operation that takes two vectors and returns a
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106:{\displaystyle \mathbf {a} \cdot \mathbf {b} =|\mathbf {a} |\,|\mathbf {b} |\cos \theta }
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822:{\displaystyle \mathbf {a} \in \mathbb {R} ^{d},\mathbf {b} \in \mathbb {R} ^{d}}
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includes a list of related items that share the same name (or similar names).
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or wedge product – a binary operation on two vectors that results in a
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is the unit vector perpendicular to the plane determined by vectors
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may refer to one of several operations between two (or more)
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exerted on a charged particle moving in a magnetic field.
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762:{\displaystyle (\mathbf {a} \otimes \mathbf {b} )}
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28:. It may concern any of the following articles:
960:Index of articles associated with the same name
353:(the area of the parallelogram formed by sides
346:{\displaystyle \mathbf {a} \times \mathbf {b} }
316:{\displaystyle \mathbf {a} \wedge \mathbf {b} }
8:
323:has the same magnitude as the cross product
401:and products between more than two vectors.
293:. In Euclidean 3-space, the wedge product
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720:{\displaystyle (a\odot b)_{i}=a_{i}b_{i}}
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574:– for two vectors, the geometric product
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907:. It can also be used to calculate the
661:– entrywise or elementwise product of
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867:– products involving three vectors.
144:{\displaystyle \mathbf {\hat {n}} }
895:occurs frequently in the study of
873:– products involving four vectors.
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988:Set index articles on mathematics
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899:, where it is used to calculate
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397:) but generalizes to arbitrary
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665:of scalar coordinates, where
390:{\displaystyle \mathbf {b} }
368:{\displaystyle \mathbf {a} }
188:{\displaystyle \mathbf {b} }
166:{\displaystyle \mathbf {a} }
854:{\displaystyle (d\times d)}
1014:
952:
557:{\displaystyle V\otimes W}
528:{\displaystyle v\otimes w}
922:done by a constant force.
918:is used to determine the
640:A bilinear product in an
948:Vector algebra relations
455:{\displaystyle w\in W,}
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426:{\displaystyle v\in V}
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993:Operations on vectors
938:Matrix multiplication
933:Scalar multiplication
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564:of the vector spaces.
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22:vector multiplication
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642:algebra over a field
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871:Quadruple products
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407:– for two vectors
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964:set index article
651:for vectors in a
568:Geometric product
495:{\displaystyle W}
475:{\displaystyle V}
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905:angular momentum
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659:Hadamard product
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572:Clifford product
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287:Exterior product
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865:Triple products
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535:belongs to the
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998:Multiplication
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537:tensor product
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508:tensor product
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405:Tensor product
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40:vector. Thus,
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971:internal link
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909:Lorentz force
906:
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893:cross product
890:
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845:
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829:results in a
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748:
732:
731:Outer product
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637:as arguments.
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504:vector spaces
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399:affine spaces
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116:Cross product
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38:
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31:
30:
29:
27:
23:
19:
881:
878:Applications
635:multivectors
536:
507:
21:
15:
916:dot product
653:Lie algebra
649:Lie bracket
33:Dot product
18:mathematics
982:Categories
843:×
805:∈
782:∈
749:⊗
679:⊙
616:∧
600:⋅
549:⊗
520:⊗
444:∈
418:∈
336:×
306:∧
266:^
256:θ
253:
208:×
136:^
101:θ
98:
53:⋅
927:See also
897:rotation
733:- where
506:, their
291:bivector
886:Physics
861:matrix.
26:vectors
969:If an
901:torque
663:tuples
462:where
120:vector
37:scalar
962:This
769:with
920:work
914:The
903:and
891:The
502:are
482:and
433:and
375:and
173:and
570:or
250:sin
95:cos
16:In
984::
647:A
195:,
20:,
849:)
846:d
840:d
837:(
815:d
810:R
801:b
797:,
792:d
787:R
778:a
757:)
753:b
745:a
741:(
727:.
713:i
709:b
703:i
699:a
695:=
690:i
686:)
682:b
676:a
673:(
655:.
644:.
620:b
612:a
608:+
604:b
596:a
592:=
588:b
583:a
552:W
546:V
523:w
517:v
490:W
470:V
450:,
447:W
441:w
421:V
415:v
384:b
362:a
340:b
332:a
310:b
302:a
272:.
263:n
246:|
241:b
236:|
230:|
225:a
220:|
216:=
212:b
204:a
182:b
160:a
133:n
91:|
86:b
81:|
75:|
70:a
65:|
61:=
57:b
49:a
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