Derivation of reciprocal vectors

Question

I know that the reciprocal vectors are defined as $v_i v_j = \delta_{ij}$, which gives ${\bf v_i} = \frac{{\bf v_j} \times {\bf v_k}}{{\bf v_i} \cdot ({\bf v_j} \times {\bf v_k})}$. However, I have been unable to find any article explaining how these were derived/obtained. Are these reciprocal vectors definitions, or is there a way of properly defining them? Thank you!

Answer 1

These are defined by $\bar{v}_i=\frac{v_j\times v_k}{v_i\bullet (v_j\times v_k)}$ and then one can easily verify that $$\bar{v}_i\bullet v_r=\frac{v_r\bullet(v_j\times v_k)}{v_i\bullet(v_j\times v_k)}$$ is equal to $\delta_{ir}$. Your conundrum is due to the fact you are not employing a different name symbol for them.