The pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies $|w1| =1$, $|w2| = 1$ and $\Re\left(w_1\overline{w_2}\right) =0$

Question

If $z_1=a + ib$ and $z_2 = c + id$ are complex numbers such that $\left|z_1\right| = \left|z_2\right| = 1$ and $\Re\left(z_1\overline{z_2})\right)=0$, then Prove that the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies $\left|w_1\right| =1$, $\left|w_2\right| = 1$ and $\Re\left(w_1\overline{w_2}\right) =0$

Answer 1

Since $w_2\bar{w}_2=|w_2|^2=1$, we get that $\mathrm{Re}(w_1\bar{w}_2)=\mathrm{Re}(w_1/w_2)=0$, and therefore, $w_1/w_2=\pm i$ since $|w_1/w_2|=1$.

So $w_1=a+ic$ can be any complex number so that $a^2+c^2=1$ and $w_2=\pm(c-ia)$.

Answer 2

The condition on $z_1, z_2$ says that $a^2 + b^2 = c^2 + d^2 = 1, $ and $z_1 = \pm i z_2.$ So, either $a = d, b = c$ or $a = -d, b = -c.$ This shows that $a^2 + c^2 = a^2 + b^2 = c^2 + d^2 = b^2 + d^2 = 1,$ and that $w_1 = \pm i w_2.$