Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\varepsilon_2$ be the set of Euclidean motions in $\mathbb{R}^2$, defined as the ordered pairs $(a,R(\phi))$ where $a \in \mathbb{R}^2$,

$$R(\phi)=\begin{pmatrix} \cos(\phi)&-\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{pmatrix} $$

and $(a,R(\phi))x=R(\phi)x+a \hspace{0.3cm} \forall x \in \mathbb{R}^2$

I'm asked to find the composition law , say $\star$, for the group. As a reference this is exercise number 3 in Balachandran Group theory and Hopf Algebras

My approach is the following:

Let $(a,R(\phi)),(b,R(\theta)) \in \varepsilon_2$ and $x \in \mathbb{R}^2$. Then

$$(a,R(\phi))(b,R(\theta))x=(a,R(\phi))(R(\theta)x+b)=(R(\phi)R(\theta)x+R(\phi)b+a) $$

so the group composition law is

$$(a,R(\phi))\star(b,R(\theta))=(R(\phi)b+a,R(\phi)R(\theta)) $$

Is this correct?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

It's correct, but you could finish by writing $R(\phi)R(\theta)=R(\phi+\theta)$ as a single rotation.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.