Vector Cross Product Distributes over Addition/Proof 3

Proof
Let $\mathbf b'$ and $\mathbf c'$ be the projections of $\mathbf b$ and $\mathbf c$ onto the plane perpendicular to $\mathbf a$.

Then $\mathbf b' + \mathbf c'$ is the projection of $\mathbf b + \mathbf c$ onto that plane.

We have:

Because:
 * $\mathbf a$ is perpendicular to $\mathbf b'$ and $\mathbf c'$
 * $\mathbf a \times \mathbf b'$ lies in the plane perpendicular to $\mathbf a$

it follows that:
 * $\mathbf a \times \mathbf b'$ is $\norm {\mathbf a}$ times the length of $\mathbf b'$ and is perpendicular to $\mathbf b'$.

Similarly:
 * $\mathbf a \times \mathbf c'$ is $\norm {\mathbf a}$ times the length of $\mathbf c'$ and is perpendicular to $\mathbf c'$.

Hence:
 * $\mathbf a \times \mathbf b' + \mathbf a \times \mathbf c'$ also lies in the plane perpendicular to $\mathbf a$.
 * $\mathbf a \times \mathbf b' + \mathbf a \times \mathbf c'$ is $\norm {\mathbf a}$ times the length of $\mathbf b' + \mathbf c'$ and is perpendicular to $\mathbf b' + \mathbf c'$.

In other words:
 * $\mathbf a \times \mathbf b + \mathbf a \times \mathbf c = \mathbf a \times \paren {\mathbf b + \mathbf c}$