Cross Product of Parallel Vectors

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Theorem

Let $\mathbf a$ and $\mathbf b$ be vector quantities which are parallel.

Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.


Then:

$\mathbf a \times \mathbf b = 0$


Proof

By definition of cross product:

$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$

where:

$\norm {\mathbf a}$ denotes the length of $\mathbf a$
$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
$\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.

By definition of parallel, the direction of $\mathbf a$ is the same as the direction of $\mathbf b$

Hence when $\mathbf a$ and $\mathbf b$ are parallel, $\theta = 0$ by definition.

Hence from Sine of Zero is Zero, $\sin \theta = 0$.

Hence the result.

$\blacksquare$


Sources

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