# Thread: 3 rotated coordinate systems.

1. ## 3 rotated coordinate systems.

I have 3 sets of X,Y,Z axes which differ by rotations only. Call them A, B, and C.
In A, there is a unit vector (0, m, n). The same unit vector relative to A exists in both B and C as (bx, by, bz) and (cx, cy, cz).
I need to find the angle between the X axis of B and the X axis of C as projected ("flattened") onto A's XY plane.

It's been 35 years since my last linear algebra class, so any clues would be greatly appreciated!

John

2. As an afterthought, the problem can be simplified.

A unit vector in X,Y,Z is (0,0.34,-0.94). (Please excuse the rounding error)

An axes rotation occurred which caused the same vector to be represented as (a,b,c).

How much did the axes rotate around the Z axis for a given a,b,c?

3. I seem to be answering my own question as things come back to me...

Here's a plan that might work:

The rotation matrix that rotates (0, .34, -.94) to the X axis is
0 , .34 , -.94
-1 , 0 , 0
0 , .94 , .34

Apply this to (a, b, c) to get the rotated X axis, X' (in terms of a,b and c).

Next find the rotation matrix T that rotates the X axis to X'

T will be the product of 3 rotations. From T, solve for the Z rotation matrix which will give the angle rotated around Z. The angle will be in terms of a,b and c.

At this point I don't know how to find T or if there is a better way.