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 particles 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.
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