Strategy 1 -- STP computation: Always set up the 3x3 determinant carefully. Expand along the row or column with the most zeros. Double-check signs in the cofactor expansion.

Strategy 2 -- Coplanarity: Set [a b c]=0 and solve for the unknown parameter. For four points, use [AB AC AD]=0. This is the fastest and most reliable test.

Strategy 3 -- VTP simplification: Always use BAC-CAB. Never try to compute the intermediate cross product first — the formula is faster and less error-prone. Remember: a x (b x c) != (a x b) x c.

Strategy 4 -- Linear combinations: If vectors are given as combinations of a base set, use [p q r]=det(coefficient matrix)*[a b c]. This avoids expanding each vector into components.

Strategy 5 -- Shortest distance: Apply the formula SD=|[a2-a1,b1,b2]|/|b1xb2| directly. The numerator is an STP (3x3 determinant) and the denominator is a cross product magnitude.

Strategy 6 -- Identities: Recognize standard identities to simplify: Jacobi (sum of cyclic VTPs = 0), Lagrange ((a x b).(c x d) = (a.c)(b.d)-(a.d)(b.c)), and [a x b, b x c, c x a]=[a b c]^2.

Common traps: (1) Forgetting the 1/6 factor for tetrahedron volume. (2) Confusing a x (b x c) with (a x b) x c. (3) Sign errors in STP (swapping two vectors flips the sign). (4) Using the skew line formula for parallel lines $\frac{gives 0}{0}$. (5) Forgetting absolute value in volume calculations. (6) Not checking whether [a b c] is zero before using reciprocal vectors.