# Angular velocity about arbitrary axis

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 $\omega_r$?

• Can you precise what you mean by the angular velocity of a point in $R$ about $\mathbf{r}$? – mathcounterexamples.net Dec 21 '17 at 9:17
• @mathcounterexamples.net You may assume any point $p=\left(p_1,p_2,p_3\right) in$R$. If, for example,$p$does not reside on the$x$-axis, it angular velocity about the$x$-axis is$\omega_x$. If$p$does not reside on$\mathbf{r}$, what is the equeivalent angulare velocity of$p$about$\mathbf{r}$. Please let me know if this is clear enough. – Asman Dec 21 '17 at 16:03 • You may assume any point$p=(p_1,p_2,p_3)$in$R$. If, for example,$p$does not reside on the$x$−axis, its angular velocity about the$x$−axis, is$\omega_x$. If$p$does not reside on$\mathbf{r}$, what is the equivalent angular velocity of$p$about$\mathbf{r}\$. Please let me know if this is clear enough. – Asman Dec 21 '17 at 16:10