group theory - How is it that angular velocities are vectors, while rotations aren't?

Does anyone have an intuitive explanation of why this is the case?

Answer

This is a note on why angular velocities are vectors, to complement Matt and David's excellent explanations of why rotations are not.

When we say something has a certain angular velocity $\vec{\omega_1}$, we mean that each part of the thing has a position-dependent velocity

$\vec{v_1}(\vec{r}) = \vec{\omega_1} \times \vec{r}$.

We might consider another one of these motions

$\vec{v_2}(\vec{r}) = \vec{\omega_2} \times \vec{r}$

and wonder what happens when we add them. We get

$\vec{v_1}(\vec{r}) + \vec{v_2}(\vec{r}) = \vec{\omega_1} \times \vec{r} + \vec{\omega_2} \times \vec{r}$.

The cross product is linear, so this is equivalent to

$(\vec{v_1} + \vec{v_2})(\vec{r}) = (\vec{\omega_1} + \vec{\omega_2}) \times \vec{r}$,

so it makes fine sense to add angular velocities by vector addition.