Q. If $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ are three non-coplanar vectors and $\overrightarrow{\mathbf{p}},\overrightarrow{\mathbf{q}},\overrightarrow{\mathbf{r}}\mathbf{,}$ are reciprocal vectors, then $(l\overrightarrow{\mathbf{a}} + m\overrightarrow{\mathbf{b}} + n\overrightarrow{\mathbf{c}}) \bullet (l\overrightarrow{\mathbf{p}} + m\overrightarrow{\mathbf{q}} + n\overrightarrow{\mathbf{r}})$ is
Explanation
(c)
$\because\ \overrightarrow{\mathbf{p}}\mathbf{,}\overrightarrow{\mathbf{q}}\mathbf{,}\overrightarrow{\mathbf{r}}$ are reciprocal vectors $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ respectively.
$\therefore\overrightarrow{\mathbf{p}}\mathbf{\bullet \ }\overrightarrow{\mathbf{a}} = 1,\overrightarrow{\mathbf{p}}\mathbf{\bullet \ }\overrightarrow{\mathbf{b}} = 0,\ \overrightarrow{\mathbf{p}}\mathbf{\bullet \ }\overrightarrow{\mathbf{c}}\mathbf{\ }$ etc.
$\therefore\left( l\overrightarrow{\mathbf{a}} + m\overrightarrow{\mathbf{b}} + n\overrightarrow{\mathbf{c}} \right) \bullet \left( l\overrightarrow{\mathbf{p}} + m\overrightarrow{\mathbf{q}} + n\overrightarrow{\mathbf{r}} \right) = l^{2} + m^{2} + n^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.