# Components of angular velocity?

Let $$\vec \omega = (\omega_1, \omega_2, \omega_3)$$ be the angular velocity of a rigid body with respect to the body frame, where the body frame is right-handed orthonormal.

I have gathered 2 definitions of $$\vec \omega$$ from different sources and I am confused at how they connect to one another. One is that the rigid body rotates with $$\vec \omega$$ through its Center of Mass at rate $$abs(\vec \omega)$$. The other is that each component of $$\vec \omega$$ represents the rate at which the rigid body rotates about that particular basis axis of the body frame.

Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis?

• @MartinCheung I am afraid that none of these answers is correct. I am not sure that the three projections of $\vec \omega$ mean that there is a rotation with $\omega_x$ around the axis $x$, with $\omega_y$ around the axis $y$, and with $\omega_z$ around the axis $z$. Jan 13, 2015 at 16:06
• @DavidHammen : Is that true that performing a rotation by an angle $\omega_x$ around the axis $x$, then by an angle $\omega_y$ around the axis $y$, and then by an angle $\omega_z$ around the axis $z$, we get in fact a rotation by $|\vec \omega|$ around the axis $\vec \omega$ ? Is that true? With the Euler angles we work differently, and as far as I know, one doesn't get the same result if one performs the rotations in a different order. Jan 13, 2015 at 18:40