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Math Help - Reflection and rotation matrices

  1. #1
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    Reflection and rotation matrices

    Let R(\theta)=\left[\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array}\right] denote the 2x2 rotation matrix.

    Show that R(\theta)R(\phi)=R(\theta+\phi)

    Without further matrix calculation, explain why R(\phi)R(\theta)=R(\theta)R(\phi)

    Anyone willing to help explain how to work this out?
    Last edited by CaptainBlack; February 8th 2009 at 12:09 PM.
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  2. #2
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    Quote Originally Posted by Konidias View Post
    Let R(\theta)=\left[\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array}\right] denote the 2x2 rotation matrix.
    Show that R(\theta)R(\phi)=R(\theta+\phi)
    Without further matrix calculation, explain why R(\phi)R(\theta)=R(\theta)R(\phi)
    Do you know how to multiply two 2x2 matrices?
    Do you know the sine and cosine of the sum of two numbers?
    That is all this problem is about.
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  3. #3
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    Quote Originally Posted by Konidias View Post
    Let R(\theta)=\left[\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array}\right] denote the 2x2 rotation matrix.

    Show that R(\theta)R(\phi)=R(\theta+\phi)

    Without further matrix calculation, explain why R(\phi)R(\theta)=R(\theta)R(\phi)

    Anyone willing to help explain how to work this out?

    The last part just requires that you observe that addition is comulative and so what you have already proven shows that:

    R(\theta)R(\phi) = R(\theta+\phi) = R(\phi+\theta)=R(\phi)R(\theta)

    CB
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    The last part just requires that you observe that addition is comulative and so what you have already proven shows that:

    R(\theta)R(\phi) = R(\theta+\phi) = R(\phi+\theta)=R(\phi)R(\theta)

    CB
    "commutative" not "comulative"!
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  5. #5
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    I forgot to ask: Show that R(\theta)R(\phi)=R(\theta+\phi) and give a geometrical interpretation of what this means.

    I understand the first bit now but how would you give a geometrical interpretation of the result?
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