Explanation

(a)
Let $A,B,\ C$ be the points represented by the numbers $\mathcal{z}_{1},\ \mathcal{z}_{2},\ \mathcal{z}_{3}\ and\ P$ be the point represented by $\mathcal{z}$
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Now, the four points $A,B,C,P$ form a parallelogram in the following three orders.
(i)$A,B,P,\ C$ (ii)$B,C,P,A$ and (iii)$C,A,P,B$
In case (i), the condition for $A,B,P,\ C$ to form a parallelogram is $\overrightarrow{\mathbf{AB}} = \overrightarrow{\mathbf{CP}}\ \ \ \ ie,$ $\mathcal{z}_{2} - \mathcal{z}_{1}\mathcal{= z -}\mathcal{z}_{3}\ \$or $\mathcal{z =}\mathcal{z}_{2} + \mathcal{z}_{3} - \mathcal{z}_{1}$
Similarly, in case (ii) and (iii), $\overrightarrow{\mathbf{BC}} = \overrightarrow{\mathbf{AP}}\ \ ie,$
$\mathcal{z}_{3} - \mathcal{z}_{2}\mathcal{= z -}\mathcal{z}_{1}\$or $\mathcal{z =}\mathcal{z}_{3} + \mathcal{z}_{1} - \mathcal{z}_{2}$
and $\overrightarrow{\mathbf{CA}} = \overrightarrow{\mathbf{BP}}$
$ie,\ \mathcal{z}_{1} - \mathcal{z}_{3}\mathcal{= z -}\mathcal{z}_{2}$
or $\mathcal{z =}\mathcal{z}_{1} + \mathcal{z}_{2} - \mathcal{z}_{3}$