Why is linear velocity represented as cross product of angular velocity of the particle and its position vector? Why not vice versa? (Consider rigid body rotation)
Answer
OK, I'm assuming you want the formal proof of this well known kinematics formula! So here goes: 
Let the particle rotate about the axis OO' ... Within time interval dt let its motion be represented by the vector dφ whose direction is along axis obeying right-hand-corkscrew rule, and whose magnitude is equal to the angle dφ.
Now, if elementary displacement of particle at a be specified by radius vector r,
From the diagram, it is easy to see that, for infinitesimal rotation, dr= dφΧr ... 1 (crossproduct)
By definition, ω = dφ/dt
Thus taking the elementary time interval as dt, all given equations surely hold!
Thus we can divide both sides of equation 1 by dt which is corresponding time interval!
So we get dr/dt = dφ/dt X r of course r value won't change WRT the particle and axis, so r/dt is essentially r!
So result is, v = ωΧr
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