Suppose $(E,M,\pi)$ is a vector bundle and $M$ can be written as a union of two open subspaces $U$ and $V$ such that $U\cap V$ is connected. Assume that the restrictions of $E$ to $U$ and $V$ are orientable. I am stuck at this problem.
Show that $E$ is orientable.
What I know about orientability:
vector bundle $(E,M,\pi)$ of rank $n$ is orientable iff $\Lambda^nE^\ast$ is isomorphic to the trivial bundle of rank $1$ over $M$.
Suppose $(E, M, \pi)$ is a vector bundle and $J:(E, M,\pi)\rightarrow (E, M,\pi)$ is a homomorphism such that $J\circ J=-id$. Then $E$ is orientable.
My initial thought was to use these above two facts to conclude $E$ is orientable. However I couldn't go further than this. Any suggestion or hints are greatly appreciated.