Scalar Triple Product Properties

Question

If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rearranged product [b c a]?

RankGuru · Accepted Answer

The scalar triple product is invariant under cyclic permutation of its vectors, meaning [a b c] = [b c a] = [c a b], while any single swap of two vectors reverses its sign. This permutation symmetry is a heavily tested JEE Advanced property rooted in determinant behavior. Moving from [a b c] to [b c a] is a cyclic shift, advancing each vector one position with a wraparound, which preserves the value entirely; therefore [b c a] = [a b c] = 7. Option -7 would arise only from a single transposition such as [b a c], which is a non-cyclic swap, not the cyclic rotation given here. Option 0 would require the vectors to be coplanar, contradicting the nonzero value 7. Option 14 doubles the product with no justification. The underlying reason is that a cyclic permutation of determinant rows corresponds to an even permutation, leaving the determinant unchanged. Plausibility check: a cyclic shift can be built from two successive row swaps, and two sign reversals multiply to +1, so the value must remain 7, consistent with the result.

Scalar Triple Product Properties

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