Question

A complex number z=3+4i is rotated about another fixed complex number $z_1=1+2i$ in anticlockwise direction by ${45}^0$ angle.Find the complex number represented by new position of z in argand plane.

Answer 1

Let us consider the problem:

$z\prime -z_{\Omega }=e^{i\theta }\left(z-z_{\Omega }\right)$

Given,

$z=3+4i$

$z_{\Omega }=1+2i$

$\theta =\frac{\pi }{4}$

$i^2=-1$

Since,

Implies that, $z\prime -\left(1+2i\right)=e^{i\frac{\pi }{4}}\left(\left(3+4i\right)-\left(1+2i\right)\right)$

Implies that, $z-z_{\Omega }=3+4i-1-2i=2+2i$

Implies that, $e^{i\frac{\pi }{4}}=\cos \left(\frac{\pi }{4}\right)+i\sin \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$

Implies that, $e^{i\frac{\pi }{4}}\left(z-z_{\Omega }\right)=\left(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}\right)\left(2+2i\right)$

Implies that, $=2\frac{\sqrt{2}}{2}\left(1+i\right)\left(1+i\right)$

Implies that, $=\sqrt{2}\left(1+2i+i^2\right)$

Implies that, $=\sqrt{2}\left(1+2i+i^2\right)$

Implies that, $=\sqrt{2}\left(1+2i-1\right)$

Implies that, $=2\sqrt{2}i$

Hence,

Implies that, $z\prime -\left(1+2i\right)=2\sqrt{2}i$

Implies that, $z\prime =2\sqrt{2}i+1+2i$

Hence, the required complex number equation represented as

$=1+i\left(2+2\sqrt{2}\right)$