Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 9 / c

Dot the reciprocal-basis expansion with each member of the original basis. All mismatched dot products vanish and the matching one equals one. ConsequentlyThese 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.