I have a robotics assignment, which I am unable to solve. Given the axis-angle rotation vector $\Theta = (2, 2, 0)$, how can I calculate the unit vector of the rotation axis $k$ and the angle $\theta$?
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Given the axis-angle rotation vector $\Theta = (2, 2, 0)$, you can find the unit vector in the same direction by diving by the norm (or length) of $\Theta$, denoted by $\|\Theta\| = \sqrt{2^2 + 2^2 + 0^2} = \sqrt{8} = 2\sqrt{2}$. Therefore, the unit vector in the direction of $\Theta$ is $k= \Theta/\|\Theta\| = (1/\sqrt{2}, 1/\sqrt{2}, 0)$, which should be the axis of rotation that you're looking for. The angle should just be the norm of $\Theta$, that is, $\theta = \|\Theta\| = 2\sqrt{2}$. Note that $k \theta$ gives you your original vector $\Theta$.
