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}\]
\[\begin{aligned}
Definition of the cross product.
\end{aligned}\]