It is difficult to design controllers for the complicated dynamics of omnidirectional vehicles steered by multiple wheels with distributed traction force. In this paper, the dynamic model of a three-wheel omnidirectional vehicle, which is linearized to simplify controller design, is developed. The conditions of making its dynamics linear are derived first. Then, a strategy of planning wheel velocities to satisfy these conditions is proposed. Consequently, three-wheel omnidirectional vehicle can be easily treated by classical linear control theories. Finally, a linear optimal tracker is designed to control the omnidirectional vehicle for desired movement trajectories. In particular, the dynamic model includes the motors installed in the three-wheel omnidirectional vehicle, making it a practical model. Three kinds of vehicle trajectories illustrate the planning of wheel trajectories for linearizing the vehicle dynamics, and simulations demonstrate the performance of the linear optimal tracker. In addition, experimental results of a practical three-wheel omnidirectional vehicle are also included.