Are Angular Quantities Vectors? — Question 8

Question 8

Mirror reflection distinguishes axial from polar vectors

A disk rotates in the $xy$ plane. At $\vec r=(R,0,0)$ a mark has velocity $\vec v=(0,V,0)$ with $R,V>0$. Reflect the physical motion in the $yz$ plane, so polar vectors transform by $x\to-x$.

1. Find the angular-velocity direction before and after reflection.

2. Explain why applying the polar-vector reflection rule directly to the original angular velocity would fail.

Question 8 - Solution

Mirror reflection distinguishes axial from polar vectors

(a) Derivation and calculation

For the original motion, $\vec v=\vec\omega\times\vec r$ gives $$(\omega_z\hat z)\times(R\hat x)=\omega_z R\hat y=V\hat y \quad\Rightarrow\quad\boxed{\vec\omega=\frac VR\hat z}.$$ After reflection, $$(\omega_z^{\prime}\hat z)\times(-R\hat x)=V\hat y \quad\Rightarrow\quad\boxed{\vec\omega^{\prime}=-\frac VR\hat z}.$$

(b) Result and interpretation

For the reflection matrix $S=\operatorname{diag}(-1,1,1)$, polar vectors obey $\vec v^{\prime}=S\vec v$. An axial vector instead obeys $$\boxed{\vec\omega^{\prime}=\det(S)S\vec\omega=-S\vec\omega}.$$ Applying only $S$ to the original $z$ vector would incorrectly leave the handedness unchanged.

Physical check

The mirrored mark moves upward at the left side of the circle, requiring clockwise rotation.