Explanation

(a)
In a parallelogram $OP_{1}P_{2}P_{3},$ the mid point of $P_{1}P_{2}\ and\ OP_{3}$ are the same. But mid point of $P_{1}P_{2}\ \ is\ \ \left( \frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2} \right)$.
So, that the coordinates of $P_{3}$ are $(x_{1} + x_{2},\ y_{1} + y_{2})$
Thus, the point $\ P_{3}$ corresponds to sum of the complex numbers $\mathcal{z}_{1}$ and $\mathcal{z}_{2}$
$\therefore\ \ {\overrightarrow{\mathbf{OP}}}_{3} = \ {\overrightarrow{\mathbf{OP}}}_{1} + {\overrightarrow{\mathbf{OP}}}_{2} = \mathcal{z}_{1} + \mathcal{z}_{2}$