  We study the evolution of a three-dimensional minisuperspace cosmological
model by the Noether symmetry approach. The phase space variables turn out to
correspond to the scale factor of a flat Friedmann-Robertson-Walker (FRW)
model, a scalar field with potential function $V(\phi)$ with which the gravity
part of the action is minimally coupled and a vector field of its kinetic
energy is coupled with the scalar field by a coupling function $f(\phi)$. Then,
the Noether symmetry of such a cosmological model is investigated by utilizing
the behavior of the corresponding Lagrangian under the infinitesimal generator
of the desired symmetry. We explicitly calculate the form of the coupling
function between the scalar and the vector fields and also the scalar field
potential function for which such symmetry exists. Finally, by means of the
corresponding Noether current, we integrate the equations of motion and obtain
exact solutions for the scale factor, scalar and vector fields. It is shown
that the resulting cosmology is an accelerated expansion universe for which its
expansion is due to the presence of the vector field in the early times, while
the scalar field is responsible of its late time expansion.
