I have a vector (<1, 0, 0>) that needs to be transformed from an initial 3d rectangular coordinate system M1 through M2 and M3 to a final 3d rectangular system M4. I'm currently doing this by applying sequential rotations omega, phi, and kappa about the x', y', and z' axes, respectively, for each individual transform, and at the end, the result is the final transformed vector. This is done through a 3x3 transformation matrix for each of M1-M4. I want to be able to combine these coordinate system rotations, though, so instead of applying M1-M4, I'm simply applying one matrix M. How is this done?
Multiply your matrices M1-M4 together, and you get M. This is valid, since the matrix product is associative.