Vector Cross Product Distributes over Addition/Proof 1

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: