I have a number of points that lie on a plane. I want to move these so that they lie on a second plane.

Do you want to keep the distances between points the same?

I know where three points on the first plane are positioned on the second plane.

How do you know these three points?

Internet searches lead me to believe that the solution may well involve rotation matrices.

With a rotation matrix you can obviously move points around. I'm not sure if this solves your problem. I think you have to supply more information on what exactly you're trying to do. Also there are many ways to handle rotation matrices, I normally look at them as a set of Euler angles but you can also use quaternions or delve into Lie algebras?!

There is a lot of SW around to handle these types of geometry problems. It's often not about the complexity of the calculations but more about understanding the particular geometry problem. I use SPICE for this. For example to project four points (corner points of a Field-of-View) from one plane onto another. This often involves a change of coordinate system as well.