# 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

$OB + BA = OA$

That is:

$\mathbf b + \vec {BA} = \mathbf a$

$\vec {BA} = \mathbf a - \mathbf b$
$\blacksquare$