Can vector addition and relative motion laws be applied to axial vectors like angular velocity, just as they are to regular vectors? Any difference? [closed]

Question

I'm having trouble understanding how to apply angular velocity in different reference frames. In my scenario, a disc is rotating with angular velocity about its center. Consider two particles on the disc, at distances and from the center. I want to calculate their relative angular velocity.

I was taught that vector addition laws apply to axial vectors, but axial vectors behave differently from normal vectors during reflections. Since both particles have the same angular velocity (assuming it's in the same direction, upwards), their relative angular velocity should be:

Wrel = W1 - W2 = 0

(because ).

However, I also know that their relative linear velocity is:

V1 - V2 = W (R1 - R2)

From this expression, if we apply (Vrel=Rrel.Wrel), it seems like , not 0. This seems contradictory. My teachers say that we can use vector addition laws for axial vectors, but they also claim that the relative angular velocity is , not 0. Here I am referring to relative angular velocity as the angular velocity of one particle with respect to the frame of the other

I'm confused about how vector laws and relative motion principles apply to axial vectors in this context. Could you explain where I'm going wrong, and how to correctly handle this using vector laws for axial vectors?

Answer 1

Angular velocities add like vectors. The relative angular velocity of the two particles is zero. You have to think of it as being in the rotating frame of one particle and looking at the other. So, if you are sitting on the disc next to one particle, going around with it, what is the apparent motion of the other particle in this rotating frame?

If the angular velocities are not identical, they still add like vectors, but the result is rather unintuitive.

Consider two aircraft flying due north at the same speed, crossing the equator at the same time, but $30^\circ$ apart in longitude. What is their relative velocity?

Intuition suggests their relative velocity should be zero, but we are talking about motion on a spherical planet, so in fact, they are both rotating about axes at right angles to the plane of the great circle they follow. The difference between these (the relative angular velocity) is a much shorter angular velocity vector pointing towards the point on the equator halfway between their paths. The relative angular velocity has each spinning about this point relative to the other. This is hard to see, so we need an animation.

In the figure below, first, watch the blue lines of latitude and longitude. You can see that the two aircraft start at one pole, move along their own meridian to the equator, separate, and then converge on the other pole. The relative motion of one aircraft, as seen in the frame of the other, forms a figure-8 shape. Note that when crossing the equator, the relative velocity is non-zero.

If we now watch the red lines, we see that at every point on its trajectory, the relative velocity is pointing parallel to the red circles. Its relative velocity is spinning around the mid-point, but the midpoint is itself moving in this rotating reference frame.

Relative angular velocity is perfectly meaningful and follows the usual vector addition rule, even when the rotation axes are not parallel. But the result is unintuitive.