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The matrices A ,(<&), >42(4>), and ;43(4>) all have the trace tr(/i (<!>))= l+2cos<J> (12-6d)

The trace of a direction cosine matrix representing a rotation by the angle $ about an arbitrary axis takes the same value. This result, which will be used without proof below, follows from the observation that the rotation matrices representing rotations by the same angle about different axes can be related by an orthogonal transformation, which leaves the trace invariant (see Appendix C).

In general, the axis of rotation will not be one of the reference axes. In terms of the unit vector along the rotation axis, รจ, and angle of rotation, the most general direction cosine matrix is

cos 4> + ef( 1 - cos 9) ete2(l- cos - e3sin $ e,e3( 1 - cos + e2sin $

e, e2( 1 - cos 9) + e3sin 4> cos $ + e2( 1 - cos e2e3( I - cos - e,sin $

et e3( 1 - cos - e2sin $ e2e3( I - cos <I>) + e,sin <l> cos$+e3(l -cos$)

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