Lagrange's identity: (a x b).(c x d) = (a.c)(b.d) - (a.d)(b.c).

Special case (a=c, b=d): |a x b|^2 = |a|^2|b|^2 - (a.b)^2. This is equivalent to |a x b| = |a||b|sin(theta).

Another useful result: [a x b, b x c, c x a] = [a b c]^2. The STP of the cross products of pairs equals the square of the original STP.

Product of two STPs: [a b c][d e f] = |a.d a.e a.f; b.d b.e b.f; c.d c.e c.f| (determinant of dot products).

These identities appear in JEE Advanced occasionally. The key ones to remember are Lagrange's identity and |a x b|^2 = |a|^2|b|^2 - (a.b)^2.