Hermitian metric induced by short exact sequence of sheaves

Question

Let $L\rightarrow X$ be a holomorphic vector bundle generated by global sections $s_1,...,s_k$. It is well known that there exists a Hermitian metric on $L$ defined in a local trivialization $\psi$ by $$||\sigma(x)||^2=\dfrac{||\psi(\sigma(x))||^2}{\sum_i||\psi(s_i(x))||^2}.$$ I saw that this can be recovered from the short exact sequence of sheaves $$0\rightarrow Ker (\varphi) \hookrightarrow \mathcal{O}_X^k \xrightarrow{\varphi} \mathcal{O}(L)\rightarrow 0,$$ where $\varphi$ is defined by $$\varphi(f_1,...,f_k)=\sum_if_is_i.$$ Can someone explain to me how to get it from that sequence?