We demonstrate a controllable p-n junction in a three-dimensional Dirac semimetal (DSM) Cd3As2 nanowire with two recessed bottom gates. The device exhibits four different conductance regimes with gate voltages, the unipolar (n-n and p-p) and bipolar (n-p and n-p) regimes, where p-n junctions are formed. The conductance in the p-n junction regimes decreases drastically when a magnetic field is applied perpendicular to the nanowire. In these regimes, the device shows quantum dot behavior, whereas the device exhibits conductance plateaus in the n-n regime at high magnetic fields. Our experiment shows that the ambipolar tunability of DSM nanowires can enable the realization of quantum devices based on quantum dots and electron optics.