{c}

Let $V=\mathbf a\cdot(\mathbf b\times\mathbf c)\ne0$. The <scalar triple product> is invariant under a cyclic permutation, so the definitions of the <reciprocal basis> give
$$
\boxed{\mathbf a'\cdot\mathbf a=\frac VV=1,\qquad\mathbf b'\cdot\mathbf b=\frac VV=1,\qquad\mathbf c'\cdot\mathbf c=\frac VV=1.}
$$

Solution

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

 Solution (source code)