In Classical Mechanics angular velocity, angular acceleration, torque and angular momentum can be defined as vectors with clear advantages such as the possibility to use vector product to simplify expressions.
As someone who appreciates the symmetry between translational and rotational dynamics writing angular velocity as the derivative of the angle seems somewhat elegant to me, however this is not accurate when using vectors. This could be solved by defining an "angle vector". Why has this not common? Wouldn't it work?
I can imagine $\vec{\theta}$ perpendicular to the plane the angle lies on and with magnitude equal to its size in radians.
Best Answer
It depends on what you mean by "vector". If by "vector" you only mean something that has a magnitude and a direction, then yes, the axis-angle representation qualifies as a "vector".
To a mathematician, a vector is something that is a member of a vector space. In this context, the axis-angle representation fails to qualify as a vector. One problem is that there are many ways to represent a null rotation (e.g., a 360 degree rotation about any axis); the null element has to be unique for a space to qualify as a vector space. Another problem is that rotations do not commute; the composition of two elements has to be commutative for a space to qualify as a vector space.
Another issue: The time derivative of the axis-angle representation is not the angular velocity. It has little physical meaning. (Similarly, the integral of the angular velocity vector has little physical meaning.)