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