1. The problem statement, all variables and given/known data Given one known orthonormal basis S in terms of the standard basis U, how would I express a third basis T in terms of U when I know its representation in S? For example, U consists of <1,0,0> <0,1,0> <0,0,1> And S consists of (for example) <0.36, 0.48, -0.8> <-0.8, 0.6, 0> <0.48, 0.64, 0.6> (Note all colums and rows have unit magnitude and are orthogonal) The values of S are in x, y, and z components of U. Now I want T in terms of U, when I know that in terms of S it is <1, 0, 0> <0, -1, 0> <0, 0, -1> I'd like to arrive at a general solution, i.e. one that works for any T (so long as it's orthonormal). 2. Relevant equations (sorry I don't know Latex) Point (in frame A) = (BtoA)RotationMatrix * Point (in frame B) 3. The attempt at a solution These bases I'm talking about are components of rigid-body frames, so each of them has an "origin", which is transformed through elementwise linear combination. (i.e. the origin of S is Sx, Sy, Sz, so add (0, 0, 0) to (Sx, Sy, Sz). Similarly, T has origin (Tx, Ty, Tz), which means that its origin in terms of U is (0+Sx+Tx, 0+Sy+Ty, 0+Sz+Tz). That is fine and good for purely translated frames, but I need now to consider rotations. T and S share the same X-direction, but the Z (hence Y to maintain right-handedness) directions are inverted. I know I can define a rotation matrix to map a single point from one frame to another (I even know how to use the 4x4 general transformation matrix to get translation&rotation in one shot) but that's not quite the problem I'm facing (I think). What I would like to do is express a direction&orientation that I know in one orthonormal basis (T known in S) in another (T unknown in U but S known in U). btw, this isn't a homework problem but I feel this forum is the most likely to generate a solution. So it's possible the problem statement needs work. Thanks for your input!