An acoustic vector-sensor (a.k.a. vector-hydrophone) is composed of three acoustic velocity-sensors, plus a collocated pressure-sensor, all collocated in space. The velocity-sensors are identical, but orthogonally oriented, each measuring a different Cartesian component of the three-dimensional particle-velocity field. This acoustic vector-sensor offers an azimuth-elevation response that is invariant with respect to the source's center frequency or bandwidth. This acoustic vector-sensor is adopted here for recursive least-squares (RLS) adaptation, to track a single mobile source, in the absence of any multipath fading and any directional interference. A formula is derived to preset the RLS forgetting factor, based on the prior knowledge of only the incident signal power, the incident source's spatial random walk variance, and the additive noise power. The work presented here further advances a multiple-forgetting-factor (MFF) version of the RLS adaptive tracking algorithm, that requires no prior knowledge of these aforementioned source statistics or noise statistics. Monte Carlo simulations demonstrate the tracking performance and computational load of the proposed algorithms.