MHB Writing formulas to describe isometries

  • Thread starter Thread starter kalish1
  • Start date Start date
  • Tags Tags
    Formulas Writing
kalish1
Messages
79
Reaction score
0
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:
Physics news on Phys.org
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).
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...
Back
Top