The homogeneous Bethe-Salpeter equation is solved in the quenched ladd
er approximation for the vector positronium states of four-component q
uantum electrodynamics in two space and one time dimensions. Fermion p
ropagator input is from a rainbow approximation Dyson-Schwinger soluti
on, with a broad range of fermion masses considered. This work is an e
xtension of earlier work on the scalar spectrum of the same model. The
nonrelativistic limit is also considered via the large fermion mass l
imit. Classification of states via their transformation properties und
er discrete parity transformations allows analogies to be, drawn with
the meson spectrum of QCD.