Vector Triple Product
Using the standard expansion identity, simplify the vector triple product a x (b x c) where a, b and c are arbitrary nonzero vectors in space.
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About This Question
- Subject
- mathematics
- Chapter
- vector algebra
- Topic
- vector triple product
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
(a.c)b−(a.b)c
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.
This medium difficulty mathematics question is from the chapter vector algebra, covering the topic of vector triple product. It appeared in the 2025 exam.
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