Let P be a point in the plane of the vectors A B =3 i + j - k and A C = i - j +3 k such that P is equidistant from the lines AB and AC. If | AP |=(√…

Let P be a point in the plane of the vectors $\displaystyle \overrightarrow{A B}=3 \hat{i}+\hat{j}-\hat{k}$ and $\displaystyle \overrightarrow{A C}=\hat{i}-\hat{j}+3 \hat{k}$ such that P is equidistant from the lines AB and AC . If $\displaystyle |\overrightarrow{\mathrm{AP}}|=\frac{\sqrt{5}}{2}$, then the area of the triangle ABP is :