  We present a detailed spectroscopic investigation of a thermal $^{87}$Rb
atomic vapour in magnetic fields up to 0.4T in the Voigt geometry. We fit
experimental spectra with our theoretical model \textit{ElecSus} and find
excellent quantitative agreement, with RMS errors of $\sim 0.3$%. We extract
the magnetic field strength and the angle between the polarisation of the light
and the magnetic field from the atomic signal and find excellent agreement to
within $\sim 1$% with a commercial Hall probe. Finally, we present an
investigation of the relative sensitivity of this technique to variations in
the field strength and angle with a view to enabling atom-based high-field
vector magnetometry.
