I am reading Hatcher's notes on vector bundles and there is the following passage considering $\mathbb{R}P^n$.
The canonical line bundle $p:E \to \mathbb{R}P^n$ has as its total space $E$ the subspace of $\mathbb{R}P^n \times \mathbb{R}^{n+1}$ consisting of pairs $(\ell,v)$ with $v \in \ell$ and $p(\ell,v)=\ell$. Local trivializations can be defined by orthogonal projection.
Now I am trying to verify these local trivializations in the case of $n=1$, but I don't quite understand how these should be given by orthogonal projections?
A local trivialization is a map $\varphi_\alpha: p^{-1}(U_\alpha) \to U_\alpha \times \mathbb{R}^2$ such that for any $x \in \mathbb{R}P^1$ and a neighborhood $U_\alpha$ of $x$ we have $p\vert_{U_\alpha} \circ \varphi_\alpha = p$ and for each $y \in U$ the restriction $\varphi_\alpha$ to $\varphi_\alpha^{-1}(y)$ is a vector space isomorphism to $\{y\}\times \mathbb{R}^2$.
So how are these actually given by orthogonal projections?
An important principle when dealing with the canonical vector bundles which come with $\mathbb{RP}^n$ (and more generally, Grassmannians over $\mathbb{R}$ and $\mathbb{C}$) is that we can write down many maps (to and) from them by writing linear maps $\mathbb{R}^{n}\to V$ (or $V\to \mathbb{R}^{n+1}$) and considering these as maps between the trivial bundle $\mathbb{RP}^{n}\times\mathbb{R}^{n+1}\to\mathbb{RP}^n\times V$. Since $E\subset \mathbb{RP}^n\times \mathbb{R}^{n+1}$, such maps produce maps $E\to \mathbb{RP}^n\times V$.
To reduce bloat, we won't consider orthogonal projection onto a subspace of $\mathbb{R}^{n+1}$, but rather simply a nonzero linear map $L: \mathbb{R}^{n+1}\to \mathbb{R}$. By the preceding paragraph, this induces a map $\hat{L}:E\to \mathbb{RP}^n\times \mathbb{R}$. If $T\in \mathbb{RP}^n$ is a subspace such that $L\vert_{T}\neq 0$, we automatically have $\hat{L}_T: E_{T}\to \mathbb{R}$ non zero (how is $\hat{L}_T$ related to $L\vert_{T}$?). Because this is a map of 1 dimensional vector spaces, it is an isomorphism wherever it is nonzero. Hence it restricts to a trivialization wherever $L\vert_{T}\neq 0$, i.e. on $\mathbb{RP}^{n}\backslash \mathbb{P}(\ker L).$ Then, if we can find a collection of linear maps $L_{\alpha}:\mathbb{R}^{n+1}\to \mathbb{R}$ such that $ \bigcap_{\alpha} \mathbb{P}(\ker L_{\alpha})=\varnothing$, we get a system of local trivializations of $E$.
Now, one can get such a family of linear maps by using the standard inner product on $\mathbb{R}^{n+1}$ and taking $L_i(u):=e_i\cdot u$ for $\{e_i\}$ a basis of $\mathbb{R}^{n+1}$