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Math Help - 3 rotated coordinate systems.

  1. #1
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    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
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  2. #2
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    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?
    Last edited by Jazzjohn; February 23rd 2010 at 02:44 PM. Reason: typo in vector
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  3. #3
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    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.
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