# Writing formulas to describe isometries

#### kalish

##### Member
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!

Last edited:

#### Deveno

##### Well-known member
MHB Math Scholar
Some thoughts to get you started on part (1):

Which points in the plane are likely candidates for the point $a$?

Which angles are likely candidates for the rotation angle $\theta$?

Try "modelling" the proposed rotation with 2 superimposed sheets of paper (semi-transparent paper might be helpful, or write very darkly on the bottom sheet so you can see it under the first sheet).