Three Dimensional Geometry: JEE Main 2025 Mathematics question

Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ be $\hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + \hat{\mathbf{k}}$, $\hat{\mathbf{i}} + 3 \hat{\mathbf{j}} - 2 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}} + \hat{\mathbf{j}} - \hat{\mathbf{k}}$ 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 $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6 \sqrt{2}}$, then the position vector of $E$ is