Advanced vector algebra builds on the dot and cross products to handle three-vector interactions through scalar and vector triple products. These tools are essential for 3D geometry, volume computation, coplanarity testing, and simplifying complex vector expressions.
The scalar triple product (STP) [a b c] = a.(b x c) computes as a 3x3 determinant with a, b, c as rows. Its absolute value equals the volume of the parallelepiped with edges a, b, c. The STP is cyclic ([a b c]=[b c a]=[c a b]), anti-symmetric under transposition ([a b c]=-[b a c]), and linear in each argument ([ka b c]=k[a b c]). The critical condition [a b c]=0 characterizes coplanarity of three vectors.
The vector triple product (VTP) a x (b x c) = (a.c)b - (a.b)c (BAC-CAB rule) produces a vector in the plane of b and c. The cross product is not associative: (a x b) x c = (a.c)b - (b.c)a lies in the plane of a and b instead. The Jacobi identity states a x (b x c) + b x (c x a) + c x (a x b) = 0.
Key applications include: volume of tetrahedron = |[a b c]|; shortest distance between skew lines = |[a2-a1, b1, b2]|/|b1 x b2|; coplanarity of four points via [AB AC AD]=0; Lagrange's identity (a x b).(c x d) = (a.c)(b.d)-(a.d)(b.c); and the identity [a x b, b x c, c x a] = [a b c]^2.
The reciprocal system of vectors (a'=(b x c)/[a b c], etc.) provides a framework for resolving vectors along non-orthogonal directions, with the property a.a'=1, a.b'=0.