Using your picture, I just called the rotational part of the velocity Vr.
I agree with your Va_ equation for the components, and also Vb_ = Va_ + Vr
For Vr, a simple calculation gives the angle of the Vr vector = 270 deg + theta
Then do your usual cos( ) and sin( ) stuff with this angle along with the magnitude to get Vr.
What sign of omega to use? The way I drew the picture for Vr and the fact that I have 270 deg as part of the angle for Vr, you use a positive value for omega. This was done so that the Vr direction would match the arrow direction of omega in the drawing. But had I used theta + 90 deg for the Vr angle and drawn Vr going up to the left instead of down to the right, then you would use a negative value for omega. Perhaps this would make more sense generally. You can take your pick. You will get the same result either way. Normally one would expect omega = theta_dot which is backwards of what the drawing has. That to me is just a point of confusion that isn't needed. I would have drawn the omega arrow the other way.
