# trouble with reflections

• Jan 14th 2010, 12:58 PM
osudude
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 $\displaystyle p_{\theta}$ is the counterclockwise rotation about (0, 0) through the angle $\displaystyle \theta$, then $\displaystyle p_{\theta} \circ R$ is the reflection $\displaystyle r_{L}$ across the line L making an angle $\displaystyle \frac{\theta}{2}$ with the positive x-axis, for any $\displaystyle \theta , 0\leq \theta \leq 2 \pi$. Thus, every reflection across a line L passing through (0, 0) equals $\displaystyle p_{\theta} \circ R$ for some angle $\displaystyle \theta$.

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

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

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

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

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??
• Jan 14th 2010, 07:31 PM
qmech
The reflection R across the x-axis is:
R(x,y) = (x,-y).