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Is there a geometrical relation between the vectors of a 3x3 orthogonal matrix and it's inverse/transpose?

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If it preserves orientation, then it's a rotation. The vectors in a matrix are the vectors to which the canonical basis is sent. In the case of a rotation, the canonical basis vectors are rotated by an angle $\theta$ around some axis, and for its inverse by an angle $-\theta$, so the vectors of both matrices are related by a rotation of $2\theta$. Were you looking for something more or is that the kind of relation you were talking about? – Vhailor Sep 7 '11 at 15:22
Yes this is what I was looking for. If M=Rx(a)*Ry(b)*Rz(c) then the inverse is M'=Rz(-c)*Ry(-b)*Rx(-a). This means that if you mirror the vectors agains yoz, then xoz, then xoy you get the inverse? Thanks – titus Sep 7 '11 at 15:30

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