  We consider photoproduction of $\omega$ mesons off complex nuclei to study
interactions of transversely and longitudinally polarized vector mesons with
nucleons. Whereas the total cross section for interactions of the transversely
polarized vector mesons with nucleons $\sigma_T=\sigma(V_TN)$ can be obtained
from coherent photoproduction, measurements of vector meson photoproduction in
the incoherent region provide a unique opportunity to extract the total cross
section for longitudinally polarized mesons interacting with nucleons
$\sigma_L=\sigma(V_LN)$, which has not yet been measured and strongly depends
on theoretical approaches. This work is stimulated by the construction of the
new experiment GlueX at Jefferson Lab, designed to study the photoproduction of
mesons in a large beam energy range up to 12 GeV.
