Let $$\overrightarrow p $$ and $$\overrightarrow q $$ be the position vectors of $$P$$ and $$Q$$ respectively, with respect to $$O$$ and $$\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.$$ The points $$R$$ and $$S$$ divide $$PQ$$ internally and externally in the ratio $$2:3$$ respectively. If $$OR$$ and $$OS$$ are perpendicular then

Let $$\alpha ,\beta ,\gamma $$ be distinct real numbers. The points with position
vectors $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$$

Let $$a, b, c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ lie in a plane, then $$c$$ is

Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$$ are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$$ then the value of the expression $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $$ is equal to