Vector Cross Product Distributes over Addition

Theorem
The vector cross product is distributive over addition.

That is, in general:


 * $\mathbf a \times \left({\mathbf b + \mathbf c}\right) = \left({\mathbf a \times \mathbf b}\right) + \left({\mathbf a \times \mathbf c}\right)$

for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.

Proof
Let:
 * $\mathbf a = \begin{bmatrix} a_x \\ a_y \\a_z \end{bmatrix}$, $\mathbf b = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$, $\mathbf c = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}$

be vectors in $\R^3$.

Then: