Show $\Bbb{R}^2/C \cong \Bbb{T} \times \Bbb{R}$ Let $\Bbb{T}$ be the multiplicative subgroup of $\Bbb{C}\setminus\{0\}$ with unit modulus and $C$ the additive subgroup of $\Bbb{R}^2$ generated by a non-zero vector. Show that $\Bbb{R}^2/C \cong \Bbb{T} \times \Bbb{R}$.
My initial thought is to in some way encorporate the first isomorphism theorem, much like the case of $\Bbb{R}/\Bbb{Z} \cong \Bbb{T}$, but I cannot quite figure it out. So another thought is to come up with some chain of isomorphisms in which each constituent isomorphism would be easier to handle. However, I have been told that there is a 'natural' isomorphism which, of course, would be preferable.
Hints and directions are appreciated.

Hint: Let $q: \mathbb{R} \to \mathbb{T}$ be the quotient map which you already know. Now you have the sequence of group homomorphisms: $$C \hookrightarrow \mathbb{R} \hookrightarrow \mathbb{R} \times \mathbb{R} \xrightarrow{q \times \operatorname{id}_{\mathbb{R}}} \mathbb{T} \times \mathbb{R}.$$ 
