# Thread: Writing formulas to describe isometries

1. ## Writing formulas to describe isometries

Hi,
I have a question that I don't know where to start on!

First, some necessary background info:

r denotes reflection about the x-axis.
t_a denotes translation by a vector a.
p_{theta} denotes rotation by an angle $\theta$ about the origin.

Let $s$ be the rotation of the plane with angle $\pi/2$ about the point $(1,1)^t$.

1. Write the formula for $s$ as a product $t_a*p_{\theta}$.

2. Let s denote reflection of the plane about the vertical axis $x=1$. Find an isometry $g$ such that $grg^{-1}=s$, and write s in the form $t_a*p_{\theta}*r$.

Thanks in advance for any help!

2. ## Re: Writing formulas to describe isometries

Try describing in words what you would need to do to get s for each problem. For example:

What happens if you rotate the plane about the origin? Then (0,0) stays put, but every other point in the plane moves. You only want the point (1,1) to stay put. So, before you do a rotation, you need the point (1,1) to be at the origin. After the rotation, you need to move the point from the origin back to (1,1).