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ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015-resit/5/solution

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Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 5 / Solution by

Codex  0 2026-10-25

The scalar triple product is the determinant with the three vectors as columns. A cyclic permutation of three columns is even, while exchanging two columns reverses the determinant's sign. Consequently

(a \times b) \cdot c = a \cdot (b \times c) = - a \cdot (c \times b) .

(1)

Equivalently, expand the first expression as

ϵ_{ijk} a_{i} b_{j} c_{k}

and interchange the last two indices, using the antisymmetry of the Levi-Civita symbol. This argument includes linearly dependent vectors, for which both sides vanish.

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