Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is