In Argand diagram,  $\mathrm{O}, \mathrm{P}, \mathrm{Q}$  represent the origin, z and z + iz respectively. then $\angle \mathrm{OPQ}=$

$$\begin{gathered} \mathrm{O}=(0, 0) , \mathrm{P}=\mathrm{z}=(\mathrm{x}, \mathrm{y}) \\ and \mathrm{Q}=\mathrm{z}+\mathrm{iz}=(\mathrm{x}+\mathrm{iy})+\mathrm{i}(\mathrm{x}+\mathrm{iy}) \\ =\mathrm{x}+\mathrm{iy}-\mathrm{y}+\mathrm{ix} \\ =(\mathrm{x}-\mathrm{y}, \mathrm{x}+\mathrm{y}) \\ \therefore \mathrm{slope} \mathrm{of} \mathrm{PQ}=\frac{\mathrm{x}+\mathrm{y}-\mathrm{y}}{\mathrm{x}-\mathrm{y}-\mathrm{x}}=\frac{-\mathrm{x}}{\mathrm{y}} \end{gathered}$$

(slope of OP)  (slope of PQ) $=\frac{\mathrm{y}}{\mathrm{x}}\times \left(\frac{-\mathrm{x}}{\mathrm{y}}\right)=-1$

$\therefore \mathrm{OP}\perp \mathrm{PQ} \Rightarrow \boxed{\overline{)\underline{\mathrm{O}}PQ}=\frac{\mathrm{\pi }}{2}}$