BTW it is true that:
wxr=v_t
where v_t=v_t theta_that=rhatxv
Since all three are now orthogonal , proof of that comes from unit vector cross product rules, basically the right hand rule anyway, except it will work for a right or left handed rule since w depends in the first place on which...
You were wrong from the start I'm afraid.
For v_t is tangential velocity (theta_hat component)
w=v_t/r
v_t=|v_t|=|rhat x v|
sometimes but not always more useful:
rhat=r/r and
v_t=|(r/r) x v|
in vector form then:
w= (rhat/r) x v=(r/r^2) x v
then
wr= rhat x v
in absolute value this is:
wr=v_t...