The counter-rotating prisms atmospheric dispersion corrector (ADC) has been widely used for the calibration of large-aperture astronomical telescopes. To achieve an optimal design method for the counter-rotating prism ADC, effectively compensate for atmospheric dispersion, and suppress the optical axis drift introduced by the ADC, we establish a vector model for ray tracing of the counter-rotating prism ADC based on traditional atmospheric dispersion compensation theory. The vector models of dispersion compensation and optical axis drift are then derived. Using this mathematical model, the impacts of different parameters of the ADCs on the dispersion compensation effect, prism rotation angle, and optical axis drift are simulated and analyzed. The simulation results show that when compensating for the same atmospheric dispersion by using the counterrotating ADC with different material combinations and bonding types, the rotation angle of the prism group remains relatively consistent, and the differences increasing as the zenith angle increases. Choosing materials with similar refractive indices near the central wavelength reduces chromatic aberration in the ADC output light and improves dispersion compensation performance. When compensating for atmospheric dispersion at different zenith angles, the offset angle of the system's optical axis decreases as the number of bonded surfaces increases. Specifically, each additional bonded surface of the optical axis drift angle can be reduced by one order of magnitude. In practical ADC design, dispersion can be effectively compensated, and optical axis drift can be suppressed by controlling the number of bonded surfaces and material selection.