A set of hyperspherical kinematic coordinates is used to describe the molecular vibrations of linear triatomic molecules. These coordinates are derived by expressing first the internal mass-scaled orthogonal vectors, which include as particular cases the Jacobi and Radau vectors, in terms of the so-called kinematic angle, which is assumed to be an optimization parameter. The magnitudes of the internal orthogonal vectors and the angle between them define a set of kinematic vibrational coordinates, which is then transformed into generalized hyperspherical coordinates. This hyperspherical transformation depends on two displacement parameters which specify the orientation and curvature of the polar variables. The resulting hyperspherical kinematic system contains, therefore, three optimization parameters. For linear triatomic molecules this system can be chosen to be a curvilinear normal system by proper selection of the displacement parameters. The utility of the hyperspherical kinematic coordinates is illustrated by carrying out vibrational calculations of vibrational energy levels for OCS and N2O. The energies obtained are also compared with the experimental values.