Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 547

As we've done in earlier chapters (see Section 5.1.3 and Section 8.4), we
describe the direction of the axis by using a unit vector n , and, as before,
the sign of n tells us which direction is considered positive rotation using
the left-hand rule. The scalar ω defines the rate of rotation, in radians per
unit time. The question we want to answer is this: What is the velocity v
of the particle at that instant?

Let's review what we already
know. First of all, from the rela-
tionship between speed and an-
gular frequency observed earlier,
we know that the speed s = v
must be ωr, where r is the radius
of the circle, or the distance be-
tween o and p . Second, v must
be perpendicular to n , or else the
particle will stray from the plane
containing the circular path, and
v must also be tangent to this
path. Thus, we know both the
magnitude and direction of the
velocity v , we just need a way
to express it algebraically. To
do so, let's introduce the vector
r = p − o , the radial vector from o to p . Note that r lies in the plane
of rotation and has a constant length, the radius of the circular path, as
shown in Figure 11.17.
Now, v is perpendicular to both r (since it's tangent to the path) and n
(since it lies in the plane of orbit). You may remember that we have a tool
that can compute a vector that is perpendicular to two other given vectors:
the cross product. Perhaps n × r = v ? The direction works out correctly, 40
but let's consider the length. Remember from Section 2.12.2 that the length
of the cross product is equal to the product of the magnitudes of the inputs,
times the sine of the angle between the two vectors. Well n is a unit vector
by assumption, and n and r are perpendicular, so the sine of the angle
between them is unity. Thus the length of the cross product n × r is simply
r . The correct speed is ωr = ω r , so we are just missing a factor of ω.

Figure 11.17
Uniform circular motion in three dimensions

40 Don't just trust Figure 11.17, use your left hand to verify this. Your thumb is
the first argument, n; index finger is the second argument, r; and middle finger is the
result, v. Look at your hand with your thumb pointing at you (it's the axis of rotation,
remember), and rotate your hand in the direction of v (your middle finger), performing
the rotation about your thumb. (The particle is at the end of your index finger.) Your
hand will rotate clockwise from your perspective, which is the definition of positive
rotation according to the left-hand rule.