For coplanarity problems: always set [a b c] = 0 and solve. This is the fastest test.

For volume problems: compute the 3x3 determinant carefully. Expand along the row with the most zeros. The tetrahedron volume is $\frac{1}{6}$ times the parallelepiped volume.

For VTP problems: always use BAC-CAB. Don't try to compute b x c first and then cross with a -- the formula is faster and less error-prone.

Remember that a x (b x c) != (a x b) x c. The JEE frequently tests this distinction.

For "simplify" type problems: look for the Jacobi identity (sum of cyclic VTPs = 0) or Lagrange's identity.

When vectors are given as linear combinations: use [p q r] = det(coefficient matrix) * [a b c]. This avoids expanding each vector.

Quick check: [a b c]^2 = [a x b, b x c, c x a]. Use this to find [a x b, b x c, c x a] without computing each cross product.

For the shortest distance between skew lines: the numerator is an STP and the denominator is |b1 x b2|. This is the most efficient formula.