During operation of many strapdown inertial navigation systems (SINS) the orientation vector of a rigid body is periodically calculated by the method of approximate solution of the Bortz approximate linear differential equation (in the practice of construction of SINS for small angles of rotation the nonlinear term in the Bortz equation is neglected). Note, that the full nonlinear Bortz equation for the vector orientation of the rigid body is an analog of the quaternion linear equation; the vector and the quaternion of the rigid body orientation are linked by known relations. In the article on the basis of the obtained exact solution of the Bortz approximate linear equation (valid for small angles of rotation of a rigid body) due to the quadratures the task is solved for determination of the quaternion orientation of a rigid body with an arbitrary angular velocity and small angle of rotation of the rigid body. Proceeding from this solution, the following approach to the construction of a new algorithm for computation of SINS orientation is proposed: 1) By the set components of the angular velocity of a rigid body on the basis of mutually - unambiguous changes of the variables at each time point, a new angular velocity of a rigid body is calculated; 2) Using the new angular velocity and the initial position of a rigid body, with the help of the quadratures we find the exact solution of the Bortz approximate linear equation (vector of orientation) with a zero initial condition; 3) The value of the quaternion orientation of a rigid body (SINS) is determined by the vector of orientation. During construction of the orientation algorithm of SINS at each subsequent step the change of the variables takes into account the previous step of the algorithm in such a way that each time the initial value of the vector orientation of a rigid body will be equal to zero.