(adsbygoogle = window.adsbygoogle || []).push({}); 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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Linear Algebra: Vector Basis (change of)

**Physics Forums | Science Articles, Homework Help, Discussion**