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Math Help - trouble with reflections

  1. #1
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    trouble with reflections

    I am having a lot of trouble understanding reflections and formulas. These two problems really have me confused. Any sort of guidance would help. Thank you.

    1)By explicit comparison of formulas, verify that if R is the reflection across the x-axis and p_{\theta} is the counterclockwise rotation about (0, 0) through the angle \theta, then p_{\theta} \circ R is the reflection r_{L} across the line L making an angle  \frac{\theta}{2} with the positive x-axis, for any  \theta , 0\leq \theta \leq 2 \pi . Thus, every reflection across a line L passing through (0, 0) equals p_{\theta} \circ R for some angle  \theta .


    so  p_{\theta}(x,y) = (xcos(\theta) - ysin(\theta), xsin(\theta) + ycos(\theta)) but what would be the formula for R so I can apply this to  p_{\theta}(R)






    2)Let R be the reflection across the x-axis and let p_{\theta} be the counterclockwise rotation about (0,0) through the angle \theta

    a) prove: R\circ p_{\theta} = p_{\theta}^{-1} \circ R

    b) prove: if  R_{L} is any reflection across a line L passing through (0, 0), then  R_{L} \circ p_{\theta} = p^{-1}_{\theta} \circ R_{L}




    a) i kinda understand this issue of inverses, am I gonna have to actually apply the formula for p theta? again, not sure about what R is.

    b) same thing, but what is the difference between R and R_L??
    Last edited by mr fantastic; January 15th 2010 at 01:47 AM. Reason: Edited title
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  2. #2
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    The reflection R across the x-axis is:
    R(x,y) = (x,-y).
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