# Thread: Determine the matrices that represent the following rotations of R^3

1. ## Determine the matrices that represent the following rotations of R^3

I need to determine the matrix that represents the following rotation of $R^3$.

(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 $(c)$?

2. ## Re: Determine the matrices that represent the following rotations of R^3

Rewriting this with the tex tags:
Originally Posted by abscissa
I need to determine the matrix that represents the following rotation of $\displaystyle R^3$.

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

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

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

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

$\displaystyle \begin{matrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{matrix}$

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

Then after simplification, the matrix is

$\displaystyle \begin{matrix} -\sqrt(3)/2 & 0 & -1/2 \\ 0 & 1 & 0 \\ 1/2 & 0 & -\sqrt(3)/2 \end{matrix}$

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