There are a number of algebraic identities involving sets of four vectors. An identity known as Lagrange's identity is given by (AxB)·(CxD) = (A·C)(B·D) - (A·D)(B·C) (Bronshtein and Semendyayev 2004, p. 185). Letting A^2 congruent A·A, a number of other useful identities include (AxB)^2 | = | A^2 B^2 - (A·B)^2 Ax(Bx(CxD)) | = | B(A·(CxD)) - (A·B)(CxD) (AxB)x(CxD) | = | (CxD)x(BxA) | = | [A, B, D] C - [A, B, C] D | = | [C, D, A] B - [C, D, B] A, where [A, B, C] denotes the scalar triple product. Equation (◇) turns out to be relevant in the computation of the point-line distance in three dimensions.

Lagrange's identity | scalar triple product | vector multiplication | vector triple product