Let $f\colon X\to S$ be a proper flat morphism of varieties, let $L_1,L_2$ be a line bundle on $X$, suppose $L_1$ and $L_2$ are fiberwise isomorphic, then See-saw lemma claims $L_1\cong L_2\otimes f^*N$ for some $N\in\mathrm{Pic}(S)$.
Suppose $L_1,L_2$ are vector bundles of the same rank, under which conditions(fiberwise) shall we have $L_1\cong L_2\otimes f^*N$ for some line bundle $N\in\mathrm{Pic}(S)$?