Vector Cross Product Operator is Bilinear

Theorem
Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ be vectors in a vector space of $3$ dimensions:


 * $\mathbf u = u_i \mathbf i + u_j \mathbf j + u_k \mathbf k$
 * $\mathbf v = v_i \mathbf i + v_j \mathbf j + v_k \mathbf k$
 * $\mathbf w = w_i \mathbf i + w_j \mathbf j + w_k \mathbf k$

where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of the vector space.

Let $c$ be a real number.

Then:
 * $\left({c \mathbf u + \mathbf v}\right) \times \mathbf w = c \left({ \mathbf u \times \mathbf w}\right) + \mathbf v \times \mathbf w$