Relative angular velocity of point with respect to another point

The relative angular velocity$~vec{omega}_{r}~$ can obtain from this equation:

$$vec{omega}_{r}=frac{vec{R}_{AB}times vec{V}_{AB} }{vec{R}_{AB}cdot vec{R}_{AB}}tag 1$$

with :

$$vec{R}_{AB}=vec{R}_{B}-vec{R}_{A}$$ $$vec{V}_{AB}=vec{V}_{B}-vec{V}_{A}$$

equation (1)

$$vec{omega}_{r}=frac{left(vec{R}_{B}-vec{R}_{A}right)times left(vec{V}_{B}-vec{V}_{A}right) }{vec{R}_{AB}cdot vec{R}_{AB}}tag 2$$

with $~vec{V}_A=vec{omega}times vec{R}_A~$ and $~vec{V}_B=vec{omega}times vec{R}_B~$

equation (2)

$$vec{omega}_{r}=frac{vec{R}_{AB}times (vec{omega}times vec{R}_{AB})}{vec{R}_{AB}cdot vec{R}_{AB}}=frac{(vec{R}_{AB}cdot vec{R}_{AB})vec{omega} - ( vec{R}_{AB}cdot vec{omega})vec{R}_{AB}}{vec{R}_{AB}cdot vec{R}_{AB}}tag 3$$

Now if A and B lie in the plane perpendicular to ω then $$vec{R}_{AB}cdot vec{omega} = vec{0}$$

equation (3) becomes:

$$vec{omega}_{r} = frac{(vec{R}_{AB}cdot vec{R}_{AB})vec{omega}}{vec{R}_{AB}cdot vec{R}_{AB}} = vec{omega}$$

thus the relative angular velocity is ω.