Geometrical Interpretation of Complex Subtraction

Theorem
Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.

Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.

Then the diagonal $BA$ of $OACB$ corresponds to $\mathbf a - \mathbf b$, the difference of $a$ and $b$ expressed as a vector.

Proof

 * Complex-Subtraction-as-Parallelogram.png

By definition of vector addition:
 * $OB + BA = OA$

That is:
 * $\mathbf b + \vec {BA} = \mathbf a$

which leads directly to:
 * $\vec {BA} = \mathbf a - \mathbf b$