Lagrange's identity connects dot products of cross products to products of dot products: (a x b).(c x d) = (a.c)(b.d) - (a.d)(b.c).

Special case (c=a, d=b): |a x b|^2 = |a|^2|b|^2 - (a.b)^2. This is the algebraic form of |a x b| = |a||b|sin(theta) and is heavily used in computation.

The identity [a x b, b x c, c x a] = [a b c]^2 relates the STP of pairwise cross products to the square of the original STP. This avoids computing three cross products individually.

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|. The product of two determinants equals the determinant of the dot product matrix. This is used when computing [a b c]^2 for vectors given in terms of magnitudes and angles rather than components.

Application: If |a|, |b|, |c| and all pairwise angles are known, [a b c]^2 can be computed from the Gram matrix without finding component representations. Example: if |a|=|b|=|c|=1 and all angles are 60 degrees, the Gram matrix has 1's on diagonal and 1/2 off-diagonal, giving [a b c]^2 = 1-3/4+2/8 = 1/2.

These identities appear in JEE Advanced problems that combine inner products with cross products. The key is recognizing which identity applies.