# Representation Of Linear Velocity as Cross Product

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)

• Are you asking why velocity is given by $v = \omega \times r$ rather than $v = r \times \omega$? – John Rennie Mar 19 '15 at 7:18
• yes!if you know the answer please let me know! – user74370 Mar 19 '15 at 7:23
• I think it's just because we conventionally take anti-clockwise angles to be positive. I can't think of any deeper reason. – John Rennie Mar 19 '15 at 7:45

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 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

• And just so u know, if u tried v = rXω, that would be fundamentally wrong, bcuz of how the direction of ω us assumed by the right hand rule convention!, then of course you would obtain an illogical result, anyway , that's if u go by convention!! – sugatasen Mar 19 '15 at 7:46
• Ok then basically i've been questioning the right hand screw rule itself..and do you know a proof for the right hand screw rule? Btw this is a good approach :-) – user74370 Mar 19 '15 at 7:56
• The right hand screw rule is just convention; we could define the cross product the opposite direction and it would be fine as long as the same new version was applied universally – danimal Mar 19 '15 at 10:27