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Concept# Seven-dimensional cross product

Summary

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 \mathbb{R}^7 a vector a × b also in \mathbb{R}^7. 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.
The product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of orthonormal basis vectors ei and ej for each i, j from 1 to 7. For example, from the table
The table can be used to calculate the product of any two vectors. For example, to calculate the e1 component of x × y the basis vectors that multiply to produce e1 can be picked out to give
This can be repeated for the other six components.
There are 480 such tables, one for each of the products satisfying the definition. This table can be summarized by the relation
where is a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365.
The top left 3 × 3 corner of this table gives the cross product in three dimensions.
The cross product on a Euclidean space V is a bilinear map from V × V to V, mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties
orthogonality:
magnitude:
where (x·y) is the Euclidean dot product and |x| is the Euclidean norm.

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Seven-dimensional cross product

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 \mathbb{R}^7 a vector a × b also in \mathbb{R}^7. 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.

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