Orthogonal splitting of a vector subbundle ($E\cong F\oplus (E/F)$)

A fiber metric on a smooth vector bundle $E$ gives a smooth orthogonal projection $P:E\to F$ onto any vector subbundle. In a local frame of $E$, let $G$ be the metric matrix and $A$ the columns of a frame of $F$. Then $P=A(A^{t}GA{)}^{-1}{A}^{t}G$, proving smoothness. The complementary subbundle $F^{\perp }=\ker P$ maps isomorphically onto the quotient vector bundle. Explicitly $v\mapsto (Pv,[v])$ is a vector bundle isomorphism $E\to F\oplus (E/F)$, with inverse $(f,[v])\mapsto f+(I-P)v$. This inverse is independent of the representative $v$, but the splitting depends on the chosen metric.