Second power of the cross product matrix
The 2nd power of the cross product matrix \[ \widetilde{V} = \left[\begin{matrix} 0 & -v_z & v_y \\ v_z & 0 & -v_x \\ -v_y & v_x & 0 \\ \end{matrix}\right] \] is \[ \widetilde{V}^2 = \left[\begin{matrix} -v_z^2 - v_y^2 & v_x v_y & v_x v_z \\ v_x v_y & -v_x^2 - v_z^2 & v_y v_z \\ v_x v_z & v_y v_z & -v_x^2 - v_y^2 \end{matrix}\right] \] Furthemore if \(v\) is unit then \[ \widetilde{V}^2 = \left[\begin{matrix} v_x^2 - 1 & v_x v_y & v_x v_z \\ v_x v_y & v_y^2 - 1 & v_y v_z \\ v_x v_z & v_y v_z & v_z^2 - 1 \end{matrix}\right] \]Proof
\[\begin{aligned}
&\widetilde{V}^2 = \widetilde{V}\widetilde{V}
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
0 & -v_z & v_y \\
v_z & 0 & -v_x \\
-v_y & v_x & 0 \\
\end{matrix}\right]
\left[\begin{matrix}
0 & -v_z & v_y \\
v_z & 0 & -v_x \\
-v_y & v_x & 0 \\
\end{matrix}\right]
\end{aligned}\]
Definition of \(\widetilde{V}\).
\[\begin{aligned}
&=
\left[\begin{matrix}
0 \cdot 0 + -v_z \cdot v_z + v_y \cdot -v_y & 0 \cdot -v_z + -v_z \cdot 0 + v_y \cdot v_x & 0 \cdot v_y + -v_z \cdot -v_x + v_y \cdot 0 \\
v_z \cdot 0 + 0 \cdot v_z + -v_x \cdot -v_y & v_z \cdot -v_z + 0 \cdot 0 + -v_x \cdot v_x & v_z \cdot v_y + 0 \cdot -v_x + -v_x \cdot 0 \\
-v_y \cdot 0 + v_x \cdot v_z + 0 \cdot -v_y & -v_y \cdot -v_z + v_x \cdot 0 + 0 \cdot v_x & -v_y \cdot v_y + v_x \cdot -v_x + 0 \cdot 0
\end{matrix}\right]
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
-v_z^2 - v_y^2 & v_x v_y & v_x v_z \\
v_x v_y & -v_x^2 - v_z^2 & v_y v_z \\
v_x v_z & v_y v_z & -v_x^2 - v_y^2
\end{matrix}\right]
\end{aligned}\]
\[\begin{aligned}
&=
\left[\begin{matrix}
v_x^2 - 1 & v_x v_y & v_x v_z \\
v_x v_y & v_y^2 - 1 & v_y v_z \\
v_x v_z & v_y v_z & v_z^2 - 1
\end{matrix}\right]
\end{aligned}\]
\(\vec{u}\) is unit. Hence
\(-v_z^2 - v_y^2 = v_x^2 - 1\),
\(-v_x^2 - v_z^2 = v_y^2 - 1\), and
\(-v_x^2 - v_y^2 = v_z^2 - 1\).