The paper presents the problem of the optimal reorientation of the spacecraft's orbit by a limited or pulse jet thrust, which is orthogonal to the plane of the osculating orbit, with the help of a quaternion differential equation of the spacecraft (SC) orbit orientation and the Pontryagin maximum principle. This kind of a jet thrust changes orientation of the spacecraft orbits while its shape and size are kept unchanged during the control process. The functional, which defines the quality of the control process, is a weighted convolution of two criteria: time and the total momentum of the jet thrust spent on the control process (special cases of this functional are the quick response problem and the problem of minimizing the characteristic velocity). In the first part of the article the authors present a review of the papers on the differential equations of the spacecraft orbit orientation and the problem of the optimal spacecraft orbit reorientation in the inertial coordinate system by a jet thrust, which is orthogonal to the plane of the osculating orbit of the spacecraft. The authors present the theory of solving the problem of the optimal reorientation of a spacecraft orbit using a quaternion differential equation of the spacecraft orbit orientation in the nonlinear continuous formulation (using a limited (small) thrust).