Let $R$ be a Cartesian right-handed frame rotating with angular velocity $\omega_x$about its $x$-axis with respect to an inertial frame $F$, that is, $\mathbf{\omega}=[\omega_x \ 0 \ 0]^T$. let's define a vector $\mathbf{r}=[r_1 \ r_2 \ r_3]^T$ in $R$.
I want to find the angular velocity $\omega_r$, that is, the angular velocity of a point in $R$ about $\mathbf{r}$ which is induced by $\mathbf{\omega}$. I would think that the answer is the inner product $\omega_r=<\mathbf{\omega},\mathbf{r}>$. However, I did some simulations, and this seems to be wrong. I think that the problem is that $\mathbf{r}$ precesses about the $x$-axis. How do I integrate the precession of $\mathbf{r}$ into an expresseion for $\omega_r$?