Let $\pi:E\to M$ be a smooth real vector bundle of rank $k$ over a smooth manifold $M$, let $g$ be a smooth fibre metric on $E$, and let $F\subset E$ be a smooth vector subbundle of rank $r$. For each $p\in M$, define
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\begin{align*}
F_p^\perp:=\{v\in E_p: g_p(v,w)=0 \text{ for every } w\in F_p\}.
\end{align*}
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Then
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\begin{align*}
F^\perp:=\bigsqcup_{p\in M}F_p^\perp\subset E
\end{align*}
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is a smooth vector subbundle of $E$ of rank $k-r$, and for every $p\in M$ there is a fibrewise direct sum decomposition