This paper presents a 3D reactive collision avoidance algorithm for vehicles with nonholonomic constraints. The algorithm steers the heading and pitch angle of the vehicle in order to maintain a constant avoidance angle to the obstacle, thus ensuring a safe collision avoidance maneuver. The flexibility provided by moving in three dimensions is utilized by choosing an optimal pair of safe pitch and heading angles for avoidance. Furthermore, the algorithm incorporates limits on the allowed pitch angle, which are often present in practical scenarios. The collision avoidance property is mathematically proved, and the analysis is validated by several numerical simulations.