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(a) angle $\theta$, the axis $e_2$

(b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$

(c) angle $\pi/2$, axis contains the vector $(1,1,0)^t$

Now, I would like to check if I got the right answers because this problem has been quite difficult for me. Any help is greatly appreciated.

Please forgive me for skipping the work because formatting matrices is a real pain. Especially when I have a lot of them.

For part $(a)$, I got that $(e_2,e_3,e_1)$ is an orthonormal basis of $R^3$. Then after simplification, the matrix is

$$

\begin{matrix}

\cos(\theta) & 0 & \sin(\theta) \\

0 & 1 & 0 \\

-\sin(\theta) & 0 & \cos(\theta) \\

\end{matrix}

$$

For part $(b)$, I got an orthonormal basis as $\{[1/\sqrt(3), 1/\sqrt(3), 1/\sqrt(3)]^t, [1/\sqrt(2),-1/\sqrt(2),0]^t,[1/\sqrt(6),1/\sqrt(6),-2/\sqrt(6)]^t\}$.

Then after simplification, the matrix is

$$

\begin{matrix}

-\sqrt(3)/2 & 0 & -1/2 \\

0 & 1 & 0 \\

1/2 & 0 & -\sqrt(3)/2 \\

\end{matrix}

$$

Is what I have done so far correct such that I can proceed with part $(c)$?