  We use heavy quark effective theory (HQET) and perturbative QCD to study the
heavy meson -- light vector meson transitions involved in D and B decays. HQET
is used to relate the measured D -> K* l nu vector and axial-vector form
factors at four-momentum transfer q^2 = 0 to the B -> K* gamma tensor and
axial-tensor form factors at q^2 = 16.5 GeV^2. Perturbative QCD is then used to
find matching conditions for the B-meson form factors at q^2 = 0. A five
parameter ``vector dominance'' type fit of the two HQET form factors,
consisting of single pole, double pole, and subtraction terms, is used to match
the data at q^2 = 16.5 GeV^2 to QCD at q^2 = 0. The values at q^2 = 0 are
compared with recent data on the exclusive rate for B -> K* gamma decay to
extract a value for the Cabibbo-Kobayashi-Maskawa matrix element Vts = 0.035,
with 28% experimental uncertainty, and 32% theoretical uncertainty from higher
order QCD effects and violations of heavy quark symmetry.
