<s>[INST] The pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies $|w1| =1$, $|w2| = 1$ and $\Re\left(w_1\overline{w_2}\right) =0$ 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 $\Re\left(w_1\overline{w_2}\right) =0$
 [/INST] 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 $w_1 = \pm i w_2.$
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