Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i +2 j + k, i +3 j -2 k and 2 i + j - k respectively. The altitude from…

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