To distinguish the low-lying vector beauty-charm meson, we systematically study the $B_c^{*}\to B_c+\gamma$, $B_c^{*}\to \ell +{\nu }_{\ell }$, and $B_c^{(*)}\to J/\psi +nh$ processes within effective theory by the helicity decomposition method. The significant difference of polarization asymmetry in $B_c^{(*)}\to J/\psi +nh$ indicates a general law in vector-to-vector and pseudoscalar-to-vector transition processes, which can be tested in current and future LHC experiments. In the end, we discuss the experiment search and discovery potential for the low-lying vector beauty-charm meson.