Vector Triple Product

The vector triple product obeys the celebrated BAC-CAB expansion identity, a x (b x c) = (a . c)b - (a . b)c, which re-expresses a nested cross product as a linear combination of the inner vectors b and c. Memorizing and correctly orienting this identity is essential for JEE Advanced problems that would otherwise require brute-force determinant expansion. The mnemonic BAC minus CAB records that the middle vector b is scaled by the dot product of the two outer vectors (a . c), while the last vector c is scaled by (a . b), with a minus sign separating them. Importantly, the result lies in the plane of b and c, since it is their linear combination, and is therefore perpendicular to b x c, which is consistent with a x (anything) being perpendicular to that anything. Option (a . b)c - (a . c)b is the negation, corresponding to (b x c) x a instead. Option (a . c)b + (a . b)c wrongly uses addition, violating the antisymmetry of the cross product. Option (b . c)a - (a . b)c misplaces a vector entirely. Plausibility check: setting a = b shows a x (a x c) = (a . c)a - |a|^2 c, the familiar special case, confirming the identity's internal consistency.