# 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 ℝ7 a vector a × b also in ℝ7. Like the cross product in three dimensions the seven-dimensional product is anticommutative and a × b is orthogonal to both a and b. Unlike in three dimensions, it does not satisfy the Jacobi identity. And while the three-dimensional cross product is unique up to a change in sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to octonions as the three-dimensional product does to quaternions.

The seven-dimensional cross product is one way of generalising the cross product to other than three dimensions, and it turns out to be the only other non-trivial bilinear product of two vectors that is vector valued, anticommutative and orthogonal. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.

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