Equations relating the velocity and angular velocity of a point in circular motion,
v = w X r (1)
w = r X v / r.r (2)
Combining these equations looking to arrive at w=w or v=v, one sees a reminder that
each is perpendicular to r.
w = r X (w X r) / r.r
v = (r X v) X r / r.r
|v| = |w| |r| sin(ph)
|w| = |r| |w| |r| sin(ph) / (|r||r|)
|w| = |w| sin(ph)
sin(ph) = 1
|w| = |r| |v| sin(th) / (|r||r|)
|v| = |r| |v| |r| sin(th) / (|r||r|)
|v| = |v| sin(th)
sin(th) = 1
By components for w,
v = w X r (1)
v1 = w2 r3 - w3 r2
v2 = w3 r1 - w1 r3
v3 = w1 r2 - w2 r1
w = r X v / r.r (2)
w1 = (r2 v3 - r3 v2)/(r.r)
w2 = (r3 v1 - r1 v3)/(r.r)
w3 = (r1 v2 - r2 v1)/(r.r)
w1 = (w1 r2 r2 - w2 r1 r2 + w1 r3 r3 - w3 r1 r3)/(r.r)
w2 = (w2 r1 r1 - w1 r1 r2 + w2 r3 r3 - w3 r2 r3)/(r.r)
w3 = (w3 r1 r1 - w1 r1 r3 + w3 r2 r2 - w2 r2 r3)/(r.r)
w1 r1 r1 = -w2 r1 r2 - w3 r1 r3
w2 r2 r2 = -w1 r1 r2 - w3 r2 r3
w3 r3 r3 = -w1 r1 r3 - w2 r2 r3
w1 r1 + w2 r2 + w3 r3 = 0 & r1 != 0
w1 r1 + w2 r2 + w3 r3 = 0 & r2 != 0
w1 r1 + w2 r2 + w3 r3 = 0 & r3 != 0
(w dot r is zero, and one or more of r1, r2, and r3 aren't)
What want for a reminder that v, w, and r be perpendicular for circular motion?
The radius is measured from the center of the circular motion under consideration.
Equation (1) happens to also be true for r measured from any point on the axis of
rotation, a common example of a convenient origin other than the center of circular
motion being the center of a rotating sphere while looking at non-equatorial points.
With Equation (2), mistaken use of a sphere center radius thusly would both reduce
the figured angular velocity magnitude and tilt its angle.