\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

Cross product matrix

The cross product \(\cross{\vec{v}}{\vec{v}}\) can be written as the multiplication of a matrix and a vector \(\widetilde{V} \vec{w}\) where \[ \widetilde{V} = \left[\begin{matrix} 0 & -v_z & v_y \\ v_z & 0 & -v_x \\ -v_y & v_x & 0 \\ \end{matrix}\right] \] \(\widetilde{w}\) is called the cross product matrix.

Proof

\[\begin{aligned} \( \widetilde{V} \vec{v} \) \end{aligned}\]
\[\begin{aligned} By definition of \(\widetilde{V}\) and \(\vec{w}\). \end{aligned}\]
\[\begin{aligned} &= \left[\begin{matrix} 0 & -v_z & v_y \\ v_z & 0 & -v_x \\ -v_y & v_x & 0 \\ \end{matrix}\right] \left[\begin{matrix} w_x \\ w_y \\ w_z \end{matrix}\right] \end{aligned}\]
\[\begin{aligned} Matrix vector multiplication. \end{aligned}\]
\[\begin{aligned} \( &= \left[\begin{matrix} 0 \cdot w_x - v_z w_y + v_y w_z \\ v_z w_x + 0 \cdot w_y - v_x w_z \\ -v_y w_x + v_x w_y - 0 \cdot w_z \end{matrix}\right] \) \end{aligned}\]
\[\begin{aligned} &= \left[\begin{matrix} v_y w_z - v_z w_y \\ v_z w_x - v_x w_z \\ v_x w_y - v_y w_x \end{matrix}\right] \end{aligned}\]
\[\begin{aligned} \(=\vec{v} \times \vec{w}\) \end{aligned}\]
\[\begin{aligned} Definition of the cross product. \end{aligned}\]