A new method has been developed, for use in magnetic confinement devices, to measure the magnetic field vector B in plasmas or gases. It utilises the intensities of the π and σ components of the resonance multiplet emitted by lithium atoms subjected to a strong Zeeman effect. A difference in dependence of these intensities on the inclination angle θ between B and the line of sight allows one to determine the direction of B, provided the intensity ratio of the π and σ components ξ(θ) is measured. The magnitude of B is routinely inferred from the width of the multiplet. The principles of the measurement are elaborated in detail for the case of a fast Li-beam (20-100 keV) used to diagnose a fusion plasma. The deviation of the population of the m- states from the statistical one due to a dominant direction for the relative velocity during the excitation of the atoms by plasma ions has been analysed and corrections to  ξ(θ) are calculated. The geometry employed for the measurement is investigated in order to minimise the uncertainties due to systematic and random errors. A procedure for in-situ calibration is outlined. As proof of the principle the results from poloidal magnetic field measurements in ohmic and H-mode pulses on the Joint European Torus (JET) are analysed. As expected, much higher components of the poloidal magnetic field BZ and BR have been found at the plasma edge in H-mode pulses indicating the sensitivity of the measurements to the bootstrap current. Reasonable agreement has been observed between the expected and obtained accuracy. The uncertainty in  ξ(θ) is found to be close to the statistical limit at  ξ(θ)> 6%. The prospects for current density measurement at the plasma edge, which remain a key issue for achieving advanced performance of modern tokamaks, are examined in terms of making use of the developed technique. It is concluded that prospects are good provided the best available Li-beam guns, with equivalent neutral current ~5mA, are used.