  We numerically study the vortex-vortex interaction in multi-component
homogeneous Bose-Einstein condensates within the realm of the Gross-Pitaevskii
theory. We provide strong evidences that pairwise vortex interaction captures
the underlying mechanisms which determine the geometric configuration of the
vortices, such as different lattices in many-vortex states, as well as the
bound vortex states with two (dimer) or three (trimer) vortices. Specifically,
we discuss and apply our theoretical approach to investigate intra- and
inter-component vortex-vortex interactions in two- and three-component
Bose-Einstein condensates, thereby shedding light on the formation of the
exotic vortex configurations. These results correlate with current experimental
efforts in multi-component Bose-Einstein condensates, and the understanding of
the role of vortex interactions in multiband superconductors.
