If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow c + {{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \over {\left| {\overrightarrow b \times \overrightarrow c } \right|}}\left( {\overrightarrow b \times \overrightarrow c } \right) = $$ ..............

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.

Evaluate $$\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $$.

A rectangle $$PQRS$$ has its side $$PQ$$ parallel to the line $$y = mx$$ and vertices $$P, Q$$ and $$S$$ on the lines $$y = a, x = b$$ and $$x = -b,$$ respectively. Find the locus of the vertex $$R$$.