Area of Parallelogram in Complex Plane

Theorem
Let $z_1$ and $z_2$ be complex numbers expressed as vectors.

Let $ABCD$ be the parallelogram formed by letting $AD = z_1$ and $AB = z_2$.

Then the area $\mathcal A$ of $ABCD$ is given by:
 * $\mathcal A = z_1 \times z_2$

where $z_1 \times z_2$ denotes the cross product of $z_1$ and $z_2$.

Proof

 * AreaOfParallelogramComplex.png

From Area of Parallelogram:
 * $\mathcal A = \text{base} \times \text{height}$

In this context:
 * $\text{base} = \cmod {z_2}$

and:
 * $\text{height} = \cmod {z_1} \sin \theta$

The result follows by definition of complex cross product.