If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a complement bundle at all? That is, a subbundle of $E$ that is pointwise a complement of $F$.
What if $F$ is of codimension $1$? Is the complement always trivial in this case?
Does anything change if $E$ is specifically the tangent bundle $TM$?