Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 1 / c / Solution

Put $V=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$. The vector triple product identity givesTaking its dot product with $\mathbf{a}\times \mathbf{b}$ and using cyclic invariance of the scalar triple product yieldsThere was no division by $V$, so this also proves the identity for coplanar vectors.