  Spin alignments of vector mesons and hyperons in relativistic heavy-ion
collisions have been proposed as signals of the global polarization. The STAR
experiment first observed the $\rm \Lambda$ polarization. Recently, the ALICE
collaboration measured the transverse momentum ($p_T$) and the collision
centrality dependence of $K^*$ and $\phi$ spin alignments in Pb-Pb collisions
at $\rm \sqrt{s_{NN}}$ = 2.76 TeV. A large signal is observed in the low $p_T$
region of mid-central collisions for $K^*$ while the signal is much smaller for
$\phi$, and these have not been understood yet. Since vector mesons have
different lifetimes and their decay products have different scattering cross
sections, they suffer from different hadronic effects. In this paper, we study
the effect of hadronic interactions on the spin alignment of $K^*$, $\phi$ and
$\rho$ mesons in relativistic heavy-ion collisions with a multi-phase transport
model. We find that hadronic scatterings lead to a deviation of the observed
spin alignment matrix element $\rho_{00}$ away from the true value for $\rho$
and $K^*$ mesons (with a bigger effect on $\rho$) while the effect is
negligible for the $\phi$ meson. The effect depends on the kinematic
acceptance: the observed $\rho_{00}$ value is lower than the true value when
the pseudorapidity ($\eta$) coverage is small while there is little effect when
the $\eta$ coverage is big. Our study thus provides valuable information to
understand the vector meson spin alignment signals observed in the experiments.
