The position vectors of the vertices $$A, B$$ and $$C$$ of a tetrahedron $$ABCD$$ are $$\widehat i + \widehat j + \widehat k,\,\widehat i$$ and $$3\widehat i\,,$$ 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 $${{2\sqrt 2 } \over 3},$$ find the position vector of the point $$E$$ for all its possible positions.

A vector $$\overrightarrow A $$ has components $${A_1},{A_2},{A_3}$$ in a right -handed rectangular Cartesian coordinate system $$oxyz.$$ The coordinate system is rotated about the $$x$$-axis through an angle $${\pi \over 2}.$$ Find the components of $$A$$ in the new coordinate system in terms of $${A_1},{A_2},{A_3}.$$

From a point $$O$$ inside a triangle $$ABC,$$ perpendiculars $$OD$$, $$OE, OF$$ are drawn to the sides $$BC, CA, AB$$ respectively. Prove that the perpendiculars from $$A, B, C$$ to the sides $$EF, FD, DE$$ are concurrent.