The cause for the preferred spin orientation in magnetic systems containing spin-1/2 transition-metal ions was explored by studying the origin of the easy-plane anisotropy of the spin-1/2 Cu(2+) ions in CuCl2·2H2O, LiCuVO4, CuCl2, and CuBr2 on the basis of density functional theory and magnetic dipole-dipole energy calculations as well as a perturbation theory treatment of the spin-orbit coupling. We find that the spin orientation observed for these spin-1/2 ions is not caused by their anisotropic spin exchange interactions, nor by their magnetic dipole-dipole interactions, but by the spin-orbit coupling associated with their crystal-field split d-states. Our study also predicts in-plane anisotropy for the Cu(2+) ions of Bi2CuO4 and Li2CuO2. The results of our investigations dispel the mistaken belief that magnetic systems with spin-1/2 ions have no magnetic anisotropy induced by spin-orbit coupling.