The construction of the correct vector material equations for nonlinear anisotropic soft magnetic materials remains one of the main reserves for increasing the accuracy of mathematical models in solving magnetostatic problems in the field formulation. The aim of the work is to establish asymptotic expressions for the reciprocity principle, which is a fundamental property of reversible magnetization processes of nonlinear anisotropic media, and to use the obtained results to optimize the computational process when constructing the vector magnetization characteristic and differential permeability tensor. The potentiality property of the magnetic flux density vector B in H-space is used. The main result of the paper is an illustration, using concrete examples, of an alternative method for calculating vector magnetization characteristics for one of the orthogonal families. In order to eliminate the instrumental error and ensure maximum accuracy and reliability of the obtained results, the exact characteristics for the components of the vector magnetization characteristic obtained by differentiating a special analytical expression for the potential were used as initial ones. The principle of reciprocity, by virtue of its universal nature, makes a significant contribution to the theory of nonlinear anisotropic media in the hysteresis-free approximation. Asymptotic expressions for the reciprocity principle are obtained for the first time. The performed computational experiments on the construction of vector characteristics based on the known magnetization characteristics in one of the directions confirm almost complete coincidence with the exact values obtained analytically. The use of asymptotic expressions for the reciprocity principle not only greatly simplifies computational processes for determining the orthogonal magnetization characteristics, but also implements the calculation of differential permeability tensors for arbitrary field values. The proposed method can be implemented in applications for calculating the magnetic field in devices with nonlinear anisotropic magnetically soft materials, primarily with cold rolled sheet electrical steels, which are most used in electrical engineering.