Pre-Quantum Electrodynamics

\begin{equation*} {\bf A} \times {\bf B} = \left| \begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array} \right| = \left( A_y B_z - A_z B_y \right) \hat{x} + \left( B_z A_x - B_x A_z \right) \hat{y} + \left( A_x B_y - A_y B_x \right) \hat{z} \end{equation*}

The cross product is distributive:

\[ {\bf A} \times ({\bf B} + {\bf C}) = {\bf A} \times {\bf B} + {\bf A} \times {\bf C} \]

The cross-product is anti-commutative:

\[ {\bf A} \times {\bf B} = - {\bf B} \times {\bf A} \]

this relation making plain that

\[ {\bf A} \times {\bf A} = 0. \]