## § Complex orthogonality in terms of projective geometry

If we think of complex vectors $p = [p_1, p_2]$, $q = [q_1, q_2]$ as belonging to projective space: that is, $p \simeq p_1/p_2$, and $q \simeq q_1 / q_2$, we can interpret orthogonality as:
\begin{aligned} p . q = 0 \\ p_1 \overline q_1 + p_2 \overline q_2 = 0 \\ p_1 / p_2 = - \overline{q_2} / \overline{q_1} \\ p = -1/\overline{q} = -q/|q| \\ \end{aligned}
If we imagine these as points on the Riemann sphere, TODO

#### § References

• Visual Complex analysis by Tristan Needham