A resolution of a coherent sheaf in terms of a full strong exceptional sequence of vector bundles

We give a locally free resolution of any coherent sheaf on a smooth projective variety which admits a full strong exceptional sequence of vector bundles. The resolution is given in terms of a full strong exceptional sequence of vector bundles. As an application, we give a new proof of classification due to Peternell-Szurek-Wi\'{s}niewski of nef vector bundles on a projective space with the first Chern class less than three and on a smooth hyperquadric with the first Chern class less than two over an algebraically closed field of characteristic zero.