= Solution
The two <reciprocal bases> satisfy $\langle u_i,w_j\rangle=\delta_{ij}$. Expanding each <vector> in its specified <basis> and using real bilinearity gives
$$
\boxed{\langle a,b\rangle=\left\langle\sum_i a_iu_i,\sum_jb_jw_j\right\rangle=\sum_i a_ib_i.}
$$
The simple sum-of-products formula holds because the two <coordinate vectors> use paired <reciprocal bases>. It does not require either <basis> to be <orthogonal>, and would generally fail if both sets of components were taken in the same nonorthogonal <basis>.
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