trouble with reflections

**osudude** · January 14th 2010, 12:58 PM

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??

**qmech** · January 14th 2010, 07:31 PM

The reflection R across the x-axis is:
R(x,y) = (x,-y).

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