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## Homework Statement

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).

## Homework Equations

(sorry I don't know Latex)

Point (in frame A) = (BtoA)RotationMatrix * Point (in frame B)

## 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!