The vector triple product a x (b x c) produces a vector, unlike the scalar triple product. The BAC-CAB rule gives: a x (b x c) = (a.c)b - (a.b)c.

Key properties: (1) The result lies in the plane of b and c (it's a linear combination of b and c). (2) The result is perpendicular to a (since it's a cross product involving a). (3) The cross product is NOT associative: (a x b) x c = (a.c)b - (b.c)a, which lies in the plane of a and b.

The BAC-CAB mnemonic: in a x (b x c), the "outside" vector a dots with the "far" vector c first, multiplied by the "middle" vector b, minus a dotted with the "middle" b times the "far" c. For (a x b) x c: the "outside" is c, and the formula is rearranged accordingly.

Jacobi identity: a x (b x c) + b x (c x a) + c x (a x b) = 0. This can be verified by expanding each VTP using BAC-CAB — all terms cancel.

Useful derived result: a x (a x b) = (a.b)a - |a|^2 b. This projects b onto the plane perpendicular to a (scaled by |a|^2). It's used to decompose a vector into components parallel and perpendicular to a given direction.

Cross product of cross products: (a x b) x (c x d) = [a b d]c - [a b c]d = [a c d]b - [b c d]a. Both expansions are valid and equal.

JEE Application: VTP problems typically ask to simplify an expression or prove an identity. Always apply BAC-CAB first; do not try to compute intermediate cross products.