question_answer

The position vectors of the vertices A, B and C of a tetrahedron ABCD are \[\hat{i}+\hat{j}+\hat{k},\] \[\hat{i}\]and 3\[\hat{i}\], respectively. The altitude from vertex D to the opposite face ABC meets the median line through A of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is \[\frac{2\sqrt{2}}{3},\] find the position vector of the point E for all its possible positions.

A)  \[\left( -1,\text{ }2,\text{ }3 \right)\text{ }or\text{ }\left( 3,-2,-1 \right)\]

B)  \[\left( -1,-3,\text{ }3 \right)\text{ }or\text{ }\left( 3,\text{ }3,\text{ }1 \right)\]

C)  \[\left( -1,\text{ }3,\text{ }3 \right)\text{ }or\text{ }\left( 3,\,-1,\,-1 \right)\]

D)  \[\left( 3,\text{ }3,-1 \right)\text{ }or\text{ }\left( -1,\text{ }3,-3 \right)\]

E)  None of these