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Math Help - Matrix

  1. #1
    Junior Member
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    Matrix

    HI, how can solved this question? What sort of eqn should i form? Right angle triangle eqn? Thanks
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  2. #2
    Junior Member roy_zhang's Avatar
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    I copied your question and the graph into the following figure, please see the attachment. consider the given matrix system, you are asking to prove that: x_2=x_1\cos\theta + y_1\sin\theta and y_2 = -x_1\sin\theta + y_1\cos\theta.

    According to your description and the attached figure, we know that x_1=OT,\;y_1=PT and x_2=PR=OQ,\;y_2=PQ. So what we need to show is OQ=(OT)\cos\theta + (PT)\sin\theta and PQ = -(OT)\sin\theta + (PT)\cos\theta. Next I am going to show these two identities actually hold, all we need is just a few manipulations of the line segments and trig. definitions. Please consider the two right triangles \triangle OST and \triangle PSQ.

    First, we have
    PQ=(PS)\cos\theta=(PT-ST)\cos\theta=\left[PT-(OS)\sin\theta\right]\cos\theta
    =\left[PT-\left(\frac{OT}{\cos\theta}\right)\sin\theta\right]\cos\theta= -(OT)\sin\theta + (PT)\cos\theta

    Then, consider
    OQ=OS+SQ=\frac{OT}{\cos\theta}+(PS)\sin\theta=\fra  c{OT}{\cos\theta}+(PT-ST)\sin\theta
    =\frac{OT}{\cos\theta}+(PT-(OT)\tan\theta)\sin\theta=\frac{OT}{\cos\theta}+(P  T)\sin\theta-(OT)\frac{\sin^2\theta}{\cos\theta}
    =(OT)\cos\theta + (PT)\sin\theta

    This completes the proof.

    Roy
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  3. #3
    Junior Member
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    With reference to the same diagram,

    If matrix A = (cos theta sin theta)
    ( -sin theta cos theta)

    1)what form of coordinate axis rotation does A^3 represent?

    2)what form of coordinate axis rotation does A^(-1) represent?
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