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Vector magnetometry was studied using the electromagnetically inducedtransparency (EIT) with linear $\perp$ linear ($lin \perp lin$) polarization ofthe probe and the pump beams in $^{87}Rb$ - $D_2$ transition. The dependence ofthe EIT on the direction of the quantization axis and the relative orientationof the polarization of the applied electric fields was studied experimentally.We have shown that from the relative strengths of the $\sigma$ and $\pi$ EITpeaks, the direction of the magnetic field can be found. Moreover from therelative separation between the $\sigma$ and $\pi$ EIT peaks, the strengths ofthe magnetic field can be calculated. We have also demonstrated that the EITpeak amplitudes show oscillatory behaviour depending upon the orientation ofthe laser polarization relative to the magnetic field direction. Using thepositions of the maxima and minima, the direction of the magnetic field can becalculated. To understand the experimental observation, a theoretical study hasbeen done numerically considering all the thirteen Zeeman sub-levels. Apartfrom the numerical model, a toy model has also been considered to obtain ananalytical response of the medium considering the velocity distribution. Thedependencies of the magnetic field direction and the polarization direction ofthe electric fields have been explicitly derived in the analytical model.Further the direction of the magnetic field is calculated using the analyticalsolution. This study can be helpful in order to make an EIT based atomic vectormagnetometer at room temperature.

Vector magnetometry using electromagnetically induced transparency with $lin\perp lin$ polarization