Complex Numbers are Parallel iff Cross Product is Zero

Theorem

Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$.

Then $z_1$ and $z_2$ are parallel if and only if:

$z_1 \times z_2 = 0$

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

Proof

By definition of complex cross product:

$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$

$\cmod {z_1}$ denotes the complex modulus of $z_1$

$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.

Necessary Condition

Let $z_1$ and $z_2$ be parallel.

Then either $\theta = 0^\circ$ or $\theta = 180^\circ$.

Either way, from Sine of Zero is Zero or Sine of Straight Angle:

$\sin \theta = 0$

and so:

$\cmod {z_1} \, \cmod {z_2} \sin \theta = 0$

Hence by definition:

$z_1 \times z_2 = 0$

$\Box$

Sufficient Condition

Let $z_1 \times z_2 = 0$.

Then by definition:

$\cmod {z_1} \, \cmod {z_2} \sin \theta = 0$

As neither $z_1 = 0$ or $z_2 = 0$ it follows that $\sin \theta = 0$.

From Sine of Multiple of Pi it follows that either:

$\theta = 0^\circ$

$\theta = 180^\circ$

or:

$\theta$ is either of the above plus a full circle.

That is, $z_1$ and $z_2$ are parallel.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Dot and Cross Product: $2.$