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

By definition of vector addition:

$OB + BA = OA$

That is:

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

which leads directly to:

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

$\blacksquare$


Sources

  • 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 3$. Geometric Representation of Complex Numbers
  • 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $8 \ \text {(a)}$
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