= Solution
Let $E(t)$ be the <matrix> whose columns are the moving <orthonormal basis> <vectors>. Then $EE^T=I$. Differentiating gives
$$
\dot EE^T+E\dot E^T=0,
$$
so $W=\dot EE^T$ is <skew-symmetric>. Every real <skew-symmetric> $3\times3$ <matrix> has the unique form
$$
W=\begin{pmatrix}0&-\omega_3&\omega_2\\\omega_3&0&-\omega_1\\-\omega_2&\omega_1&0\end{pmatrix},
$$
which acts on a <vector> as $Wv=\boldsymbol\omega\times v$. Since $\dot E=WE$, each column satisfies
$$
\boxed{\dot{\mathbf e}_i=\boldsymbol\omega\times\mathbf e_i.}
$$
The <vector> $\boldsymbol\omega$ is the instantaneous <angular velocity>. It may also be written $\boldsymbol\omega=\tfrac12\sum_i\mathbf e_i\times\dot{\mathbf e}_i$.
Back to article page
