Let $M$ be a smooth paracompact manifold, let $\pi:E\to M$ be a smooth real vector bundle of rank $r\geq 1$, and let $\det E:=\Lambda^r E$ denote its determinant line bundle. Then $E$ is orientable if and only if $\det E$ admits a nowhere-vanishing smooth section $\sigma:M\to \det E$.
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Equivalently, orientations of $E$ are in bijection with nowhere-vanishing smooth sections of $\det E$ modulo multiplication by positive smooth functions $f:M\to \mathbb{R}_{+}$.