Is quotient bundle isomorphic to the orthogonal complement? Let $p:T\longrightarrow B$ be a vector bundle with $p|_E:E\longrightarrow B$ its subbundle. We can consider the quotient bundle of $p|_E:E\longrightarrow B$ which is defined by introducing an equivalence relation on fibers of $T$ identifying vectors which difference lies in the subspace $E_b\subseteq T_b$. Denote this bundle by $T/E\longrightarrow B$. 
On the other hand, assuming that $T$ is given an inner product, there is a bundle $E^{\perp}\longrightarrow B$ being the orthogonal complement to $E$.
Are $T/E$ and $E^{\perp}$ isomorphic vector bundles?

Yes, the restriction of the quotient map $T\rightarrow T/E$ to $E^{\perp}$ induces an isomorphism $E^{\perp}\rightarrow T/E$.
