Orthogonal complex vectors with same magnitude of each entry Given a complex vector $z = (z_1, \ldots, z_n)$, how to find a complex vector $y = (y_1, \ldots, y_n)$ such that $z \cdot y = 0$ (with respect to the complex inner product) and $|y_i| = |z_i|$ for each $i$? Is this always possible? If not, then what conditions does $z$ have to satisfy for this to be possible?
It is not always possible to find $y$ such that $z \cdot y = 0$. For example, if $z=(z_1, 0, \cdots, 0)$, where $|z_1| \neq 0$ then $y = (|z_1|e^{i \theta},0,\cdots,0)$, which implies $$z.y = |z_1|^2 e^{i \phi} \neq 0.$$
The dot product is zero when the 'vectors' are normal. Hence, choose $y=ze^{\pm i\pi/2}=\pm iz$.