Canonical bundle and Möbius bundle

Question

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} \times \mathbb{R}^2 \, | \ v \perp l \}$ and $p(l,v)=v$ and the Moebius bundle $\eta:=\{E,q,S^1\}$ where $E:=\frac{[0,1] \times \mathbb{R}}{(0,x)\sim(1,-x)}$. How can I prove this bundles-isomorphism?