Question

If $z_1=a+ib$ and $z_2=c+id$ are complex number such that $\left|z_1\right|=\left|z_2\right|=1$ and Re $\left(z_1{\overline{z}}_2\right)=0,$ then show that the pair of complex numbers $w_1=a+ic$ and $w_2=b+id$ satisfies the following
(i) $\left|w_1\right|=1$
(ii) $\left|w_2\right|=1$
(iii) Re $\left(w_1{\overline{w}}_2\right)=0$

Answer 1

$a=\cos \theta ,b=\sin \theta$
$c=\cos \varphi ,d=\sin \varphi$
Re$\left(z_1{\overline{z}}_2\right)=0$
$\Rightarrow$ $0-\varphi =\frac{n\pi }{2},n=\pm 1$
$\Rightarrow$ $c=\sin \theta ,d=-\cos \theta$
$\Rightarrow$ $w_1=\cos \theta +i\sin \theta$
$w_2=\sin \theta -i\cos \theta$
$\Rightarrow$ $\left|w_1\right|=1,\left|w_2\right|=1$
$w_1{\overline{w}}_2=\cos \theta \sin \theta -\sin \theta \cos \theta +i({\sin }^2\theta -{\cos }^2\theta )=-i\cos 2\theta$
$\Rightarrow$ Re$\left(w_1{\overline{w}}_2\right)=0$