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 $$\overrightarrow p $$ 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

The vector $$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$ is

If the vectors $$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$ are not coplanar, then prove that the vector
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) + \left( {\overrightarrow a \times \overrightarrow c } \right) \times \left( {\overrightarrow d \times \overrightarrow b } \right) + \left( {\overrightarrow a \times \overrightarrow d } \right) \times \left( {\overrightarrow b \times \overrightarrow c } \right)$$ is parallel to $$\overrightarrow a .$$