Crossvector

Crossvector is a term used in some mathematical contexts to denote a cross product–like binary operation on vectors. Given two vectors a and b in a real Euclidean space, the crossvector c = a × b is a vector in the same space that is perpendicular to both a and b, with magnitude |c| = |a||b| sin θ, where θ is the angle between a and b. The operation is bilinear and alternating: a × a = 0 and a × b = − b × a. In three dimensions, the crossvector reduces to the ordinary cross product.

In seven dimensions, there exists a crossvector defined through the imaginary parts of octonion multiplication that satisfies similar orthogonality and magnitude properties; its existence is tied to the exceptional algebraic structure of octonions. In other dimensions, a binary crossvector with all the same properties does not exist (in the nontrivial sense).

Construction in three dimensions can be expressed via the Levi-Civita symbol, with components c_i = ε_ijk a_j b_k. In seven dimensions, the crossvector can be constructed using the octonion algebra, where a and b are treated as purely imaginary octonions and c = Im(a b). These constructions share the key features of orthogonality to the inputs, a magnitude given by the parallelogram area, and an antisymmetric, bilinear nature.

Applications of the three-dimensional crossvector include torque, angular momentum, and orientation calculations in physics and computer graphics. In higher dimensions, the crossvector concept is primarily of theoretical interest, informing studies of rotations, normed division algebras, and exceptional algebraic structures.

See also: cross product, vector product, octonions.