Two dimensional euclidean motions

Question

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?

Answer 1

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