Angular Velocity is Vector Quantity

Theorem
The physical quantity that is angular velocity can be correctly handled as a vector.

Proof
In order to show that angular velocity is a vector, it is sufficient to demonstrate that it fulfils the vector space axioms.

Specifically, all we need to do is demonstrate the following.

Let $\bsomega_1$ be the angular velocity of a body about an axis which passes through a fixed point $O$.

Let $P$ be a point in $B$ whose position vector with respect to $O$ is $\mathbf r$.

The tangential velocity of $P$ is $\bsomega_1 \times \mathbf r$.

Now let $\bsomega_2$ be the angular velocity about a different axis which also passes through $O$ at an angle to the first axis.

The tangential velocity of $P$ due to this angular velocity is $\bsomega_2 \times \mathbf r$.

But linear velocities combine as vectors.

Hence the total linear velocity of $P$ is given by:

Hence the motion of $P$ is the same as it would be due to an angular velocity which is the vector sum of the two components.