Scalar and vector product of two vectorsHard

Position Vector of Altitude Intersection

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $\hat i + 2\hat j + \hat k$ , $\hat i + 3\hat j - 2\hat k$ and $2\hat i + \hat j - \hat k$ 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

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