Vector Cross Product Calculator

Cross product as a vector × vector → vector only works in 3D and 7D (by Hurwitz theorem). In higher dimensions, use the wedge product (exterior algebra) or generalized cross products. The 3D cross product is special because it relates to the Lie algebra of rotations.

a × b is the 'determinant' of [i j k; a₁ a₂ a₃; b₁ b₂ b₃] expanded along the first row. The scalar triple product a·(b×c) = det[a;b;c] gives the volume of a parallelepiped.

When a and b are parallel (or one is zero). Since ||a × b|| = ||a||||b||sin(θ), the cross product is zero iff θ = 0 or π (parallel or anti-parallel). This is the test for collinearity in 3D.