  We describe a method to dispersively detect all three vector components of an
external magnetic field using alkali atoms based on the Voigt effect. Our
method relies on measuring the linear birefringence of the radio frequency
dressed atomic medium via polarization homodyning. This gives rise to modulated
polarization signals at the first and second harmonic of the dressing
frequency. The vector components of the external magnetic field are mapped onto
the quadratures of these harmonics. We find that our scheme can be utilised in
both cold and hot atomic gases to detect such external fields in shielded and
unshielded environments. In the shielded hot vapour case we achieve field
sensitivities well below 1~pT$/\sqrt{\textrm{Hz}}$ range for all 3 vector
components, using pump-probe cycles with 125 Hz repetition rate, and limited by
the short coherence time of the cell. Finally, our scheme has a simple single
axis beam geometry making it advantageous for miniature magnetic field sensors.
