# 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? Thanks!

• Can you tell me what means the word "wrt"? – Sofia Jan 13 '15 at 9:19
• @Sofia "With respect to" I guess. – Steeven Jan 13 '15 at 9:20
• @Steeven : thank you. Now, since you understood the question, what are w = (w1, w2, w3) ? Projections of the angular velocity? The text is quite badly formulated, e.g. "the RB rotates about w". Probably he means that the body rotates about an axis parallel with w. Do I guess correctly? – Sofia Jan 13 '15 at 9:25
• @Steeven : by the way, right now landed a -1 on this question. I've seen more naïve questions than this. – Sofia Jan 13 '15 at 9:26
• Not -1 but 0 as of now; I've offset the downvote. This is not a naive question. It is a good question. BTW @Sofia, wrt always means "with respect to", and wlog always means "without loss of generality". – David Hammen Jan 13 '15 at 9:47

• @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 $z$. – Sofia Jan 13 '15 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. – Sofia Jan 13 '15 at 18:40