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Math Help - Determine the matrices that represent the following rotations of R^3

  1. #1
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    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 $(c)$?
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Determine the matrices that represent the following rotations of R^3

    Rewriting this with the tex tags:
    Quote Originally Posted by abscissa View Post
    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)?
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