  We study rank 3 stable bundles E on P^3 as extensions of a line bundle B on a
smooth surface S in P^3 by the direct sum of three copies of O_{P^3}(-\nu). In
most cases, S (the dependency locus of three sections of E(\nu)) lies in the
Noether-Lefschetz locus. We give a detailed analysis when S contains a line L
and B is constructed from divisors of the form aL+bC for H=L+C a hyperplane
section of S. We study the parameter space of this construction and compare it
to the full (Gieseker-Maruyama) moduli space. We also analyse the case when B
is a power of the hyperplane bundle. The same approach is used to study rank 2
bundles on P^3.
