We present an approach for computing the heading direction of a vehicle by processing measurements from a 2-axis magnetometer rapidly. The proposed method relies on a non-linear transformation of the measurement data comprising only two inner products. Deterministic analysis of the signal model shows how the heading direction is contained in the signal and the proposed estimator is analyzed in terms of its statistical properties. Experimental verification indicates that good performance is achieved under the presence of saturation, measurement noise, and near field effects.