# Given an axis-angle rotation vector, how can I find the unit rotation axis and angle?

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 $$\theta$$?

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 $$\Theta$$.