Let $X$ be a smooth, projective variety and $\pi:E \to X$ be a projective bundle of rank $r$ i.e., each fiber of $\pi$ is a projective space of dimension $r$. Let $F \subset E$ be a projective sub-bundle of $E$, over $X$ of codimension $1$ i.e., the fiber over each point of $\pi|_F$ is of dimension $r-1$. Then,
1) Is the pull-back of the tautological line bundle $\mathcal{O}_E(-1)$ to $F$, the tautological line bundle $\mathcal{O}_F(-1)$?
2) Is the line bundle associated to $F$, $\mathcal{O}_E(-F)$ isomorphic to the tautological line bundle $\mathcal{O}_E(-1)$?