# 3 rotated coordinate systems.

• Feb 22nd 2010, 12:45 PM
Jazzjohn
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!(Thinking)

John
• Feb 23rd 2010, 09:29 AM
Jazzjohn
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?
• Feb 23rd 2010, 04:00 PM
Jazzjohn
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.