Q. Let $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ be three non-coplanar vectors and $\overrightarrow{\mathbf{r}}$ be any vector in space such that $\overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} = 1,\ \overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 2\ and\mathbf{\ }\overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{c}} = 3$. If $\left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack = 1$, then $\overrightarrow{\mathbf{r}}$ is equal to
A) $\overrightarrow{\mathbf{a}}\mathbf{+}2\overrightarrow{\mathbf{b}}\mathbf{+}3\overrightarrow{\mathbf{c}}$
B) $\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}\mathbf{+}2\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\mathbf{+}3\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}$
C) $\left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{\ } \right)\overrightarrow{\mathbf{a}}\mathbf{+}2\left( \overrightarrow{\mathbf{c}}\mathbf{\bullet}\overrightarrow{\mathbf{a}}\mathbf{\ } \right)\overrightarrow{\mathbf{b}}\mathbf{+}3\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\mathbf{+}\overrightarrow{\mathbf{c}}$
D) None of these
Explanation
(b)
Let $\overrightarrow{\mathbf{r}} = l\left( \overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} \right) + m\left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right) + n(\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}})$
$\overrightarrow{\mathbf{r}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} = l\left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack$
$\Rightarrow l = 1$
Similarly, $m = 2,\ n = 3$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.