Question

Show that the area of the triangle on the argand diagram formed by the complex number $z,iz$ and $z+iz$ is $\frac{1}{2}|z{|}^2$

Answer 1

We have, $iz=z{e}^{i\pi /2}$. This implies that $iz$ is the vector obtained by rotating vector $z$ in anti-clockwise direction through $90^{\circ }$. Therefore, $OA\perp AB$. So,

Area of $\Delta OAB=\frac{1}{2}OA\times OB$
$=\frac{1}{2}|z||iz|=\frac{1}{2}|z{|}^2$