Direction of rotation in transformation

Question

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?

Answer 1

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