Vector Cross Product Operator is Bilinear

Theorem

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

\(\ds \mathbf u\)

\(=\)

\(\ds u_i \mathbf i + u_j \mathbf j + u_k \mathbf k\)

\(\ds \mathbf v\)

\(=\)

\(\ds v_i \mathbf i + v_j \mathbf j + v_k \mathbf k\)

\(\ds \mathbf w\)

\(=\)

\(\ds w_i \mathbf i + w_j \mathbf j + w_k \mathbf k\)

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

Let $c$ be a real number.

Then:

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

Proof

\(\ds \left({c \mathbf u + \mathbf v}\right) \times \mathbf w\)

\(=\)

\(\ds \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ c u_i + v_i & c u_j + v_j & c u_k + v_k \\ w_i & w_j & w_k \end{vmatrix}\)

Definition of Vector Cross Product

\(\ds \)

\(=\)

\(\ds \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ c u_i & c u_j & c u_k \\ w_i & w_j & w_k \end{vmatrix} + \begin{vmatrix} \mathbf i& \mathbf j & \mathbf k \\ v_i & v_j & v_k \\ w_i & w_j & w_k \end{vmatrix}\)

Determinant as Sum of Determinants

\(\ds \)

\(=\)

\(\ds c \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ u_i & u_j & u_k \\ w_i & w_j & w_k \end{vmatrix} + \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ v_i & v_j & v_k \\ w_i & w_j & w_k \end{vmatrix}\)

Determinant with Row Multiplied by Constant

\(\ds \)

\(=\)

\(\ds c \left({\mathbf u \times \mathbf w}\right) + \mathbf v \times \mathbf w\)

Definition of Vector Cross Product

$\blacksquare$