The NMR shift arising from the electron angular momentum and the electron spin dipolar-nuclear spin angular momentum interaction has been examined for a $3d_1$ system in a strong octahedral crystal field when the threefold axis is chosen as the quantization axis. To investigate the NMR shift in this situation, first, we have extended the evaluation of the hyperfine integrals to any pairs of 3d orbitals adopting a general method which is applicable to a general vector R, pointing in arbitrary direction in space. Secondly, a general expression using a nonmultipole technique is derived for the NMR shift resulting from the electron angular momentum and the electron spin dipolar-nuclear spin angular momentum interactions. From this expression all the multipolar terms are determined. ${Delta}B/B$ for the $3d_1$ system in this case is compared with that for the 3d1 system when the z axis is chosen as the quantization axis. When we choose the threefold axis as the quantization axis, it is found that along the , and axes, ${Delta}B/B$ values are significantly different from each other and along the , <-1-1-1>, <-11-1>, , <-1-11>, , and <-111> axes, ${Delta}B/B$ values are however the same. We also find that the 1/R7 term contributes dominantly to the NMR shift for all values of R. When 1/$R^5$ term is included, there is good agreement between the exact solution and the multipolar terms when $R; {leqslant};0.35;nm.$