 Solution

With the chosen basis, let ${\mathbf{b}}_1=(u,v)$. The reciprocal lattice conditions giveso $u=v=\pi /a$. Similarly, ${\mathbf{a}}_1\cdot {\mathbf{b}}_2=0$ and ${\mathbf{a}}_2\cdot {\mathbf{b}}_2=2\pi$ giveAll reciprocal points areThus, in coordinates measured in $\pi /a$, the two integer coordinates have the same parity. The reciprocal lattice is another rotated square Bravais lattice, with nearest-neighbor spacing $\sqrt{2}\pi /a$.