Solution
= Solution
For the mutually orthogonal direct <primitive lattice translation vectors> $\mathbf a_i$, define the <reciprocal lattice> basis by
$$
\boxed{\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij},\qquad \mathbf b_i=\frac{2\pi\mathbf a_i}{|\mathbf a_i|^2}.}
$$
Thus $\mathbf G=\sum_i h_i\mathbf b_i$, with integer $h_i$, satisfies $\exp(i\mathbf G\cdot\mathbf R)=1$ for every <lattice vector> $\mathbf R$.