Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 548

Putting this all together, we have the formula for the velocity of a
particle with radial vector r = p − o rotating about the axis n at an
angular rate of ω radians per unit time:

Calculating linear point
velocity from angular
velocity

v = ω n × r .

As we discussed in Section 8.4, angular velocity is often described in expo-
nential map form by a single vector ω = ω n (note the boldface ω to indicate
a vector quantity). In this case, the formula is even simpler.

Calculating Linear Point Velocity from Angular Velocity

v = ω × r . (11.31)

Now let's consider the opposite problem. Assume we know p and v , and
we wish the measure the angular velocity relative to o . Again, we can use
the cross product, but this time, we need a division to get the right length:

Angular velocity of a
particle relative to an
arbitrary point

ω = r × v

r 2 .

(11.32)

To understand the division by r 2 , consider two points on a rigid disk
that is rotating around its center. Assume that angular velocity is measured
relative to this center. One point has the radial vector r , and another point
has a radial vector k r , which is in the same direction from the center, but at
a distance scaled by the factor k. These two points (indeed, all the points on
the disk) should have the same angular velocity. Thus one division by r
is necessary to compensate for the change in r as we adjust the radius. The
extra division is necessary because the outer points have a higher velocity;
if we move on the disk by scaling r by k, the new point will have a velocity
that also is scaled by k.
Although thus far we have been assuming that p is actually rotating
about o , it may not be. It might be rotating about some other point,
or moving in a straight line. However, we can still calculate the angular
velocity of p relative to o . Essentially, what Equation (11.32) tells us is
what the angular velocity would be if p were indeed orbiting around o , in
the plane containing both r and v . The axis of rotation, which is parallel
to ω, is perpendicular to this plane. Actually, there is one slight wrinkle— r
and v might not be perpendicular, which, of course, they would be if the
particle were orbiting around o . The cross product in Equation (11.32)