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If we rotate the set of axes in counter-clockwise through 3 Euler's angles to get the transformation matrix, then what about the direction of rotation to get direct transformation instead of the previously performed 3 transformations? Is the direction of rotation of set of axis in direct transformation clockwise or counter-clockwise?

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  • $\begingroup$ In any case you try to find the resulting rotation (here or the composition of two rotations for example) what you find in the end is the quantity $\:\mathbf{n} \sin\theta\:$. Then you have two options : (1) to choose $\:\mathbf{n}\:$ as direction and $\:\theta\:$ as rotation angle or (2) to choose $\:\mathbf{n'}=-\mathbf{n}\:$ as direction and $\:\theta'=-\theta\:$ as rotation angle. Of course $\:\mathbf{n'} \sin\theta'=\mathbf{n} \sin\theta$. $\endgroup$ – Frobenius Sep 10 '17 at 21:07
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    $\begingroup$ Take a look in my answer therein : Euler Rotations in Ordinary Space. From equation (03) you will understand better ja72's answer. $\endgroup$ – Frobenius Sep 10 '17 at 21:23
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You have the 3×3 rotation matrix and you want to get the axis and angle that corresponds to the identical rotation.

Look up Determining the axis angle of a rotation. The matrix trace is used to find the angle $${\rm tr}(R) = 1 + 2 \cos \theta\;\Longrightarrow\; \theta = \cos^{-1}\left( \frac{{\rm tr}(R)-1}{2} \right)$$

The the axis is $${\bf n} = \frac{1}{2\sin\theta} \pmatrix{ R_{3,2}-R_{2,3} \\ R_{1,3}-R_{3,1} \\ R_{2,1}-R_{1,2} }$$

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