Jacobi identity: a x (b x c) + b x (c x a) + c x (a x b) = 0. Each term is a VTP; their sum always vanishes.

Simplifying complex cross products: Any expression involving nested cross products can be expanded using BAC-CAB. Example: (a x b) x (c x d) = [a b d]c - [a b c]d (treating a x b as a single vector and applying the rule).

Lagrange's identity: (a x b).(c x d) = (a.c)(b.d) - (a.d)(b.c). This is derived by writing (a x b).(c x d) = a.[b x (c x d)] and expanding the VTP.

Proving vector identities: The VTP expansion is the primary tool for proving most vector product identities in the JEE syllabus.