Q. Let $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\ and\ \overrightarrow{\mathbf{c}}$ be three non-coplanar vectors and let $\overrightarrow{\mathbf{p}}\mathbf{,}\overrightarrow{\mathbf{q}}\ and\ \overrightarrow{\mathbf{r}}$ be vector defined by the relations. $\overrightarrow{\mathbf{p}}\mathbf{=}\frac{\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}\mathbf{,}\overrightarrow{\mathbf{q}}\mathbf{=}\frac{\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}}{\mathbf{\lbrack}\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}and\ \overrightarrow{\mathbf{r}}\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}}{\mathbf{\lbrack}\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}$. Then, the value of the expression$\left( \overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{+}\left( \overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{+ (}\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{a}}\mathbf{) \bullet}\overrightarrow{\mathbf{r}}$ is equal to
Explanation
(d)
Given, $\overrightarrow{a},\overrightarrow{b}\ ,\ \overrightarrow{c}$ are three non-coplanar vectors and $\overrightarrow{p},\overrightarrow{q}\ ,\ \overrightarrow{r}$ defined by the relations
$\overrightarrow{\mathbf{p}}\mathbf{=}\frac{\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}\mathbf{,\ }\overrightarrow{\mathbf{q}}\mathbf{=}\frac{\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}$ and $\overrightarrow{\mathbf{r}}\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}$
$\therefore\ \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\bullet (}\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}\mathbf{)}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}} = 1$
and $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{\bullet}\frac{\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}}\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\bullet (}\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\mathbf{)}}{\mathbf{\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack}} = 0$
Similarly, $\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{=}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{r}} = 1$
and $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{r}}\mathbf{=}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{=}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{=}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{=}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{r}} = 0$
$\therefore\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{+}\left( \overrightarrow{\mathbf{b}}\mathbf{+}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{+ (}\overrightarrow{\mathbf{c}}\mathbf{+}\overrightarrow{\mathbf{a}}\mathbf{) \bullet}\overrightarrow{\mathbf{r}}$
$= \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{p}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{q}}\mathbf{+}\overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{r}}\mathbf{+}\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{r}}$
$= 1 + 1 + 1 = 3$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.