  We present the first spin alignment measurements for the $K^{*0}(892)$ and
$\phi(1020)$ vector mesons produced at mid-rapidity with transverse momenta up
to 5 GeV/c at $\sqrt{s_{NN}}$ = 200 GeV at RHIC. The diagonal spin density
matrix elements with respect to the reaction plane in Au+Au collisions are
$\rho_{00}$ = 0.32 $\pm$ 0.04 (stat) $\pm$ 0.09 (syst) for the $K^{*0}$
($0.8<p_T<5.0$ GeV/c) and $\rho_{00}$ = 0.34 $\pm$ 0.02 (stat) $\pm$ 0.03
(syst) for the $\phi$ ($0.4<p_T<5.0$ GeV/c), and are constant with transverse
momentum and collision centrality. The data are consistent with the unpolarized
expectation of 1/3 and thus no evidence is found for the transfer of the
orbital angular momentum of the colliding system to the vector meson spins.
Spin alignments for $K^{*0}$ and $\phi$ in Au+Au collisions were also measured
with respect to the particle's production plane. The $\phi$ result, $\rho_{00}$
= 0.41 $\pm$ 0.02 (stat) $\pm$ 0.04 (syst), is consistent with that in p+p
collisions, $\rho_{00}$ = 0.39 $\pm$ 0.03 (stat) $\pm$ 0.06 (syst), also
measured in this work. The measurements thus constrain the possible size of
polarization phenomena in the production dynamics of vector mesons.
