About: Seven-dimensional cross product

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In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector a × b also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions.

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rdfs:label	Produit vectoriel en dimension 7 (fr) Seven-dimensional cross product (en) Produto cruzado de sete dimensões (pt) Векторное произведение в семимерном пространстве (ru)
rdfs:comment	En mathématiques, et plus précisément en algèbre linéaire, le produit vectoriel en dimension 7 est une loi de composition interne d'un espace euclidien à 7 dimensions, ayant certaines propriétés du produit vectoriel usuel (en dimension 3) ; on démontre d'ailleurs que de telles lois n'existent qu'en dimensions trois et sept. (fr) Em matemática, o produto cruzado de sete dimensões é uma operação bilinear em vetores em euclidianos. Ele atribui a quaisquer dois vetores, a, b em R7 um vector a × b também em R7. (pt) In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector a × b also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. (en) В математике, векторное произведение в семимерном пространстве — билинейная операция над векторами в . Оно назначает для любых двух векторов a, b в 7 вектор a × b 7. В семи измерениях существует и векторное произведение с участием шести векторов (которое является линейным, но не бинарным). (ru)
foaf:depiction
dcterms:subject	Operations on vectors Bilinear maps Octonions
Wikipage page ID	2368531 (xsd:integer)
Wikipage revision ID	1123986756 (xsd:integer)
Link from a Wikipage to another Wikipage	Quaternion Jacobi identity Cross product Mathematics SO(3) Norm (mathematics) Geometric algebra Gramian matrix Modular arithmetic Operations on vectors Hodge dual Orthogonality Angle Anticommutativity Lie algebra Commentarii Mathematici Helvetici Composition algebra Orthonormal basis Malcev algebra G2 (mathematics) Alternating operator Euclidean space Parallelepiped Fano plane Hurwitz's theorem (normed division algebras) Bilinear map Bilinear maps Bivector Octonions Dot product Associator Octonion Orthogonal complement Orthogonal group Pseudoscalar Exterior product Parallelogram Scalar triple product Vector triple product Seven-dimensional space Vector (mathematics) Bilinear operation Normed division algebra Multi-linear Trivector
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=kOsybQWDK4oC
sameAs	Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product Seven-dimensional cross product
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thumbnail	wiki-commons:Special:FilePath/Fano_plane_for_7-D_cross_product.svg?width=300
has abstract	In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector a × b also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results. (en) En mathématiques, et plus précisément en algèbre linéaire, le produit vectoriel en dimension 7 est une loi de composition interne d'un espace euclidien à 7 dimensions, ayant certaines propriétés du produit vectoriel usuel (en dimension 3) ; on démontre d'ailleurs que de telles lois n'existent qu'en dimensions trois et sept. (fr) Em matemática, o produto cruzado de sete dimensões é uma operação bilinear em vetores em euclidianos. Ele atribui a quaisquer dois vetores, a, b em R7 um vector a × b também em R7. (pt) В математике, векторное произведение в семимерном пространстве — билинейная операция над векторами в . Оно назначает для любых двух векторов a, b в 7 вектор a × b 7. В семи измерениях существует и векторное произведение с участием шести векторов (которое является линейным, но не бинарным). Как и более привычное трёхмерное векторное произведение, бинарное векторное произведение в семи измерениях является кососимметричным и ортогональным к исходным векторам, но, в отличие от того случая, оно, однако, не удовлетворяет тождеству Якоби. Семимерное векторное произведение имеет такое же отношение к октонионам как трёхмерное векторное произведение к кватернионам, и можно показать, что кроме тривиальных случаев нуля и одного измерений, бинарное векторное произведение существует только в трёх и семи измерениях. (ru)
gold:hypernym	Operation

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