Let $S$ be any quasi-compact scheme and $\mathcal{E}$ be a locally free sheaf of rank $r$ on $S$ and $X = \mathbb{P}(\mathcal{E})$ be its associated projective bundle with structure map $f : X \rightarrow S$. I want to show that why the below complex is exact sequence of vector bundles on $X$: $ 0 \rightarrow \mathcal{O}_{X}(-r) \otimes \bigwedge^{r} f^{\ast}(\mathcal{E}) \rightarrow ... \rightarrow \mathcal{O}_{X}(-1) \otimes f^{\ast}(\mathcal{E}) \rightarrow \mathcal{O}_{X} \rightarrow 0 $.
I know that it is complex because locally it is just koszul complex but I don't know why it is exact. Is it somehow the complex obtained from regular sequences? Any help would be great.