The coagulation of dust particles immersed in plasmas is studied in the presence of a relative drift between dust and plasma species. The plasma environment is responsible for the deposition of a net charge on the dust particles; the relative drift causes an anisotropy of the species deposition on the dust surface and leads to a dipole moment. The charge and dipole moment affect the coagulation rate significantly. In particular, the dipole moment is responsible for a coagulation rate dependent upon the orientation of the coagulating particles with respect to the dipole moment. This effect may be responsible for the linear structures observed in agglomerates of dust particles formed by coagulation in experimental plasma reactors.