Q. If $\overrightarrow{\mathbf{p}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}},\overrightarrow{\mathbf{q}} = 4\widehat{\mathbf{k}} - \widehat{\dot{\mathbf{j}}}\ and\ \overrightarrow{\mathbf{r}} = \widehat{\dot{\mathbf{i}}} + \widehat{\mathbf{k}},$then the unit vector in the direction of $3\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} - 2\overrightarrow{\mathbf{r}}$ is
A) $\frac{1}{3}\left( \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right)$
B) $\frac{1}{3}\left( \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} - 2\widehat{\mathbf{k}} \right)$
C) $\frac{1}{3}\left( \widehat{\dot{\mathbf{i}}} - 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}} \right)$
D) $\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
Explanation
(a)
$3\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} - 2\overrightarrow{\mathbf{r}} = 3\left( \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} \right) + \left( 4\widehat{\mathbf{k}} - \widehat{\dot{\mathbf{j}}} \right) - 2\left( \widehat{\dot{\mathbf{i}}} + \widehat{\mathbf{k}} \right)$
$= \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$\therefore\ \ Unit\ vector\ in\ the\ direction\ of\ \ 3\overrightarrow{\mathbf{p}}$+$\overrightarrow{\mathbf{q}} - 2\overrightarrow{\mathbf{r}}$
$= \frac{1}{3}(\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.