We report a novel quantification method of tri-axial magnetic anisotropy in orthorhombic substances containing rare earth (RE) ions using tri-axial magnetic alignment and tri-axial magnetic anisotropies depending on the type of RE in RE-based cuprate superconductors. From the changes in the axes for magnetization in magnetically aligned powders of (RE1-x'REx '')(2)Ba4Cu7Oy [(RE', RE '') 247] containing RE ions with different single-ion magnetic anisotropies, the ratios of three-dimensional magnetic anisotropies between RE'247 and RE '' 247 could be determined. The results in (Y, Er) 247, (Dy, Er) 247, (Ho, Er) 247, and (Y, Eu)247 systems suggest that magnetic anisotropies largely depended on the type of RE ' (or RE ''), even in the heavy RE ions with higher magnetic anisotropies. An appropriate choice of RE ions in RE-based cuprate superconductors enables the reduction of the required magnetic field for the production of their bulks and thick films based on the tri-axial magnetic alignment technique using modulated rotation magnetic fields. (C) 2014 AIP Publishing LLC.