We present a detailed spectroscopic investigation of a thermal 87 Rb atomic vapour in magnetic fields up to 0.4 T in the Voigt geometry. We fit experimental spectra with our theoretical model ElecSus and find excellent quantitative agreement, with RMS errors of ∼ 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 ∼ 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.