The application of tomographic techniques for the study of vector fields has been demonstrated only in the last few years. We show its potential for remote sensing of plasma vector fields such as the fluid velocity v and the magnetic field B using Doppler and Zeeman spectroscopy. Simulations suggest the possibility of time-resolved reconstruction of ion velocity flow fields and poloidal magnetic fields from direct measurements of the low-order spectral moments of line-integrated spectroscopic measurements. Other possible applications of vector tomography are examined.