Estimates of the three-dimensional ion velocity field can be difficult to make with monostatic radars because there are three unknown components for each independent line of sight velocity measurement. To cope with this problem, one or more assumptions about the vector field must be made to arrive at a solution. At Arecibo, one can measure the ion vector velocities by continuously rotating the antenna beam back and forth 360 degrees in azimuth at 15 degrees off zenith to sample the horizontal components. Until recently, the line of sight velocities obtained from this experiment were typically converted into vector velocities by assuming that the vector field remains constant during one rotation and that horizontal gradients are negligible. In this paper we show how to apply the linear regularization inversion method to the problem of computing ion vector velocities. This technique improves the accuracy of the vector velocities obtained from the measured F region line of sight velocities and also provides a convenient way to do the computations for dual-beam experiments. The technique could improve the vector velocities at other monostatic incoherent scatter radar facilities.