= Solution
Write $\Delta=\langle\mathbf u,\mathbf v,\mathbf w\rangle=\mathbf u\cdot(\mathbf v\times\mathbf w)$. The displayed definitions require $\Delta\ne0$, equivalently <linear independence> of the original <vectors>. The <vector triple product identity> gives
$$
\begin{aligned}
\mathbf V\times\mathbf W
&=\frac{(\mathbf w\times\mathbf u)\times(\mathbf u\times\mathbf v)}{\Delta^2}\\
&=\frac{\mathbf u\bigl((\mathbf w\times\mathbf u)\cdot\mathbf v\bigr)
-\mathbf v\bigl((\mathbf w\times\mathbf u)\cdot\mathbf u\bigr)}{\Delta^2}
=\frac{\mathbf u}{\Delta}.
\end{aligned}
$$
Here the first <scalar triple product> is $\Delta$ by cyclic permutation, and the second is zero because a <cross product> is <orthogonal> to both its factors. Thus
$$
\boxed{\langle\mathbf U,\mathbf V,\mathbf W\rangle
=\frac{(\mathbf v\times\mathbf w)\cdot\mathbf u}{\Delta^2}
=\frac1\Delta.}
$$
This is the <reciprocal basis involution> identity. When $\Delta=0$, the proposed <reciprocal basis> is undefined; for example, three <vectors> in the $xy$ <plane> cannot be inserted into the printed quotients.
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