I Rigid Body Rotation relative velocity question

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This is a question about the concepts behind rigid body rotation when we use relative velocity.

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 vA and vB respectively. These velocities are in random directions.

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 vA components. Thus the rotation component will be 'fixed' at point A and will have a relative velocity of B with respect to A, BvA, acting at point B. Why is the relative velocity vector (BvA) necessarily normal/perpendicular to the line between A and B?

My thoughts are the following: if we have randomly chosen the velocity vectors of A and B, how can we assume that BvA = vB - vA is perpendicular to BrA?

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.
 

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if we have randomly chosen the velocity vectors of A and B, how can we assume that BvA = vB - vA is perpendicular to BrA?
The rigidity of the body imposes an additional constraint on vA and vB - they must be such that the distance between points A and B does not change. That additional constraint is sufficient to justify this assumption.
(so it's not really an assumption, it's a provable consequence of the assumption of rigidity)
 
The rigidity of the body imposes an additional constraint on vA and vB - they must be such that the distance between points A and B does not change. That additional constraint is sufficient to justify this assumption.
(so it's not really an assumption, it's a provable consequence of the assumption of rigidity)
Thank you very much for your response- it makes sense now.
 

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