Question

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

Answer 1

For a rotation of center $\Omega$ and angle $\theta ,$ we have

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

Here, $z=3+4i$ and $z_{\Omega }=1+2i$ and $\theta =\frac{\pi }{4}$

$\therefore z^{\prime }-(1+2i)=e^{\frac{i\pi }{4}}((3+4i)-(1+2i))$

$\Rightarrow z-z_{\Omega }=3+4i-1-2i=2+2i$

$e^{\frac{i\pi }{4}}=\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}$

$=\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}$

$\Rightarrow e^{\frac{i\pi }{4}}(z-z_{\Omega })=(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})(2+2i)$

$=2\times \frac{\sqrt{2}}{2}(1+i)(1+i)$

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

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

$=2\sqrt{2}i$

Finally, $z^{\prime }-(1+2i)=2\sqrt{2}i$

$\Rightarrow z^{\prime }=2\sqrt{2}i+1+2i$

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