= Solution
Dot the reciprocal-basis expansion with each member of the original <basis>. All mismatched dot products vanish and the matching one equals one. Consequently
$$
\boxed{\alpha=\mathbf r\cdot\mathbf a,\qquad
\beta=\mathbf r\cdot\mathbf b,\qquad
\gamma=\mathbf r\cdot\mathbf c.}
$$
These are coefficients in the <reciprocal basis>. They would instead involve dot products with the reciprocal vectors if $\mathbf r$ were being expanded in the original <basis>. The two coefficient conventions should not be interchanged for a nonorthonormal <basis>.
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