 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/1/solution

In components, the scalar triple product is

a \cdot (b \times c) = a_{1} (b_{2} c_{3} - b_{3} c_{2}) + a_{2} (b_{3} c_{1} - b_{1} c_{3}) + a_{3} (b_{1} c_{2} - b_{2} c_{1}) .

It is the determinant with rows

a, b, c

. A cyclic permutation of the rows has positive sign, so

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

. Equivalently the Levi-Civita symbol expression

ϵ_{ijk} a_{i} b_{j} c_{k}

is unchanged by that cyclic relabeling.

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