Suppose $X_{1},X_{2}$ are smooth complex projective varieties, with rank $2$ vector bundles $E_{i} \rightarrow X_{i}$. Suppose furthermore $X_{1}$ is birational to $X_{2}$.
Question: Is $\mathbb{P}(E_{1})$ birational to $\mathbb{P}(E_{2})$?
If so is there a way to "see" the birational map between these varieties?
$(k=\mathbb{C}).$