Let $\pi:E\to M$ be a rank-$k$ real vector bundle over a smooth manifold $M$, with $k\in\mathbb N$. Let $\{(U_i,\Phi_i)\}_{i\in I}$ be a bundle atlas, where
for every $x\in U_i\cap U_j$ and $v\in\mathbb R^k$. Then $E$ is orientable if and only if $E$ admits an equivalent bundle atlas $\{(V_a,\Psi_a)\}_{a\in A}$ whose transition maps $h_{ab}:V_a\cap V_b\to GL(k,\mathbb R)$ satisfy
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\begin{align*}
\det h_{ab}(x)>0
\end{align*}
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for every $a,b\in A$ and every $x\in V_a\cap V_b$.