I am trying to visualize the following rotation of $\mathbb R^3$, but it is very difficult. I want to get the answer by intuition, and not by using the Rodrigues rotation formula or conjugation of matrices, etc.
Help please.
Problem statement: Determine the matrix that represents the following rotation of $\mathbb R^3$: an angle of $\pi/2$ about the fixed axis containing the vector $(1,1,0)^t$
Here is what I have tried in my diagram:
Should I find a 3x3 rotation matrix $A$ such that $A(1,1,0)^t=(-1,1,0)^t$?
By working out the problem using the conjugate matrices, I end up getting the answer to be:
A = \begin{pmatrix} 0.5 & 0.5 & 0.5 & \\ 0.5 & 0.5 & -\sqrt(2)/2\\ -\sqrt(2)/2 & \sqrt(2)/2 & 0 \end{pmatrix}
This answer is not at all intuitive.
If you are rotating around an axis, all points on that axis will remain the same after transformation. (e.g. If you rotate around $(1,1,0)^t$, $\alpha(1,1,0)^t = A \alpha(1,1,0)^t ,\forall \alpha \in \Bbb R$)
It may help to start by thinking of rotation about a different axis. ($\hat y$, for example) What happens to the points upon rotation around this axis? Which change? Which remain the same? Can you create a transformation matrix for this simpler case?