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 $\AA$ of $ABCD$ is given by:

$\AA = z_1 \times z_2$

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

Proof

From Area of Parallelogram:

$\AA = \text{base} \times \text{height}$

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.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Dot and Cross Product: $4.$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Dot and Cross Product: $41$