Determining angular velocity radius and velocity of the center of rotation

Hey,

I am trying to solve the following problem:

v_a=v_r+ ω × r_ra
v_b=v_r+ ω ×(r_ra+r_ab)
v_c=v_r+ ω × (r_ra+r_ac)

I know the vectors va, vb, vc, r_ab and r_ac and I want to know the angular velocity vector, the radius r_ra and the velocity v_r. Physically it means that I know 3 points on a rigid body and I want to decompose their velocities in an angular and linear component by determing around which point they rotate.

So there are 9 equations and 9 unkowns. But if I try to solve this problem analytically I can not come to any answer. I also put it in MATLAB but this also gives weird results.
My first attempt was subtracting the vectors which eventually gave me a solution for omega. But than I still could not solve the problem.

Does anyone know why this is the case? Is it something with the outer product which I don't understand? Are there infinite number of solution to this problem? Or do I just do something wrong?

Any help would be appreciated since I am already trying things for some days.

Thanks.