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In general, let us say that we have a rigid body and on it are two points, A and B, which are moving with velocities

**v**and

_{A}**v**respectively. These velocities are in random directions.

_{B}The theory states that we can view the effect of these velocities as the sum of a translation and a rotation. I am comfortable with the above and understand it.

My question lies in the nature of the rotation component. Let us say that our translation consisted of the

**v**components. Thus the rotation component will be 'fixed' at point A and will have a relative velocity of B with respect to A

_{A}**,**

**, acting at point B. Why is the relative velocity vector (**

_{B}v_{A}**) necessarily normal/perpendicular to the line between A and B?**

_{B}v_{A}My thoughts are the following: if we have randomly chosen the velocity vectors of A and B, how can we assume that

**=**

_{B}v_{A}**v**-

_{B}**v**is perpendicular to

_{A}**?**

_{B}r_{A}The explanations in the textbook are slightly contradictory in my opinion. In one case, they say that: [itex] \textbf{relative v} = \omega \times \textbf{r} [/itex], but they also seem to construct vector diagrams of velocity vectors (v_a, v_b, and b_v_a)- the latter approach makes no sense in that the direction of the relative velocity vector is set by the velocity vectors, which could be random.

Thanks for reading this and any clarification is greatly appreciated.