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Tractor Beam

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I'm implementing something that is like a "tractor beam". I've got it pretty much done but I'm have a bit of trouble turning the basis vectors into a rotation matrix (or a quaternion) to apply to the object being "tractored" so that it stays perpendicular to the tractor beam (that's a requirement).


What I have available:


I know the start position of the tractor beam.


I know where it hits my target (intersection) which is an OBB


I know the  where it hits my target on the plane facing the source. This gives me a vector of the tractor beam.



I compute the normal to the plane and get the angular difference between the the tractor beam and the normal to the plane 


thetaZ = dot_product(plane_normal,tractor_beam) / (L2norm(plane_normal) * L2norm(tractor_beam));


angleZRadians = acos(thetaZ);


make a long story short----I can compute the angle differences for all three basis vectors.....


So my question is, how do I convert this to a rotation matrix I can apply to my target, to keep it perpendicular to the tractor beam?


For instance, the tractor beam is Z axis and the  source rotates about the z axis, my computations give me deltas for x and y (say 20 degrees)

and if I try to create a ZYX order rotation I need to rotate X or Y about Z but not both. It seems to me there should be a cleaner way to get a matrix given the difference in angles of two bases. I guess another way to say it is how to create a rotation matrix given two known poses.





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Well I might have found my own answer, but if someone would confirm that would be great.


It seems I need to take the vectors of the new basis and put them as column vectors in matrix M. Then take the Inverse (transpose in this case) and use this matrix. I wonder if this would need to be a Quaternion since my tractor beam source can move in a complete orbit around the target?

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