I Angular Velocity: Vector or Not?

[e2m2a]

I understand that angular velocity is technically not a vector so does that mean the cross product of the radius vector and the angular velocity vector, the tangential vector, is also not a vector?

 

[vanhees71]

Angular velocity is an axial vector, i.e., it transforms as $\vec{\omega} \rightarrow \vec{\omega}$ under space reflections, while a polar vector like the usual velocity transforms as $\vec{v} \rightarrow -\vec{v}$. Wrt. rotations axial vectors transform in the same as polar vectors.

An axial vector is always equivalent to an antisymmetric tensor. For Cartesian components you can map the axial-vector components to the tensor components via
$$\Omega_{jk}=\epsilon_{jkl} \omega_l$$
and the other way
$$\omega_l=\frac{1}{2} \epsilon_{jkl} \Omega_{jk}.$$
In both formulae the Einstein summation convention is used, and $\epsilon_{jkl}$ is the Levi-Civita symbol which is defined by $\epsilon_{123}=1$ and being antisymmetric under exchange of any index pair. This implies that it's 0 if at least two of the indices are equal.

It's also easy to see that the cross product of two polar or two axial vectors is an axial vector, while the cross product of a polar and axial vector is a polar vector.

 

[jbriggs444]

I am told that angular velocity is not a vector so does that mean the cross product between the radius vector and angular velocity vector, the tangential velocity vector, is also not a vector?

If you are working in the two dimensional plane then angular velocity has only one dimension. It is a scalar.

If you are working in three dimensions then angular velocity is a pseudovector. It has magnitude and direction, but if you do a mirror reflection on your coordinate system so that all of your vectors change signs, the pseudovectors remain unchanged. So picky mathematical types do not want to call them vectors.

Taking the cross product of the angular momentum [pseudo]vector and the radius vector will do just fine to get you a velocity vector.

 

[Dale]

In three dimensions, the cross product of two (polar) vectors gives a pseudovector. The cross product of a pseudovector and a vector gives a vector. The cross product of two pseudovectors gives a pseudovector.

 

[e2m2a]

If you are working in the two dimensional plane then angular velocity has only one dimension. It is a scalar.

If you are working in three dimensions then angular velocity is a pseudovector. It has magnitude and direction, but if you do a mirror reflection on your coordinate system so that all of your vectors change signs, the pseudovectors remain unchanged. So picky mathematical types do not want to call them vectors.

Taking the cross product of the angular momentum [pseudo]vector and the radius vector will do just fine to get you a velocity vector.

Ok. Thanks for the explanation.

 

[e2m2a]

In three dimensions, the cross product of two (polar) vectors gives a pseudovector. The cross product of a pseudovector and a vector gives a vector. The cross product of two pseudovectors gives a pseudovector.

All right. Thanks. That clears things up.