Q. The line through $\widehat{i} + 3\widehat{j} + 2\widehat{k}$ and $\bot$ to the line $\overrightarrow{r} = \left( \widehat{i} + 2\widehat{j} - \widehat{k} \right) + \lambda(2\widehat{i} + \widehat{j} + \widehat{k})$ and $\overrightarrow{r} = \left( 2\widehat{i} + 6\widehat{j} + \widehat{k} \right) + \mu(\widehat{i} + 2\widehat{j} + 3\widehat{k})$ is

A) $\overrightarrow{r} = \left( \widehat{i} + 2\widehat{j} - \widehat{k} \right) + \lambda( - \widehat{i} + 5\widehat{j} - 3\widehat{k})$
B) $\overrightarrow{r} = \widehat{i} + 3\widehat{j} + 2\widehat{k} + \lambda(\widehat{i} - 5\widehat{j} + 3\widehat{k})$
C) $\overrightarrow{r} = \widehat{i} + 3\widehat{j} + 2\widehat{k} + \lambda(\widehat{i} + 5\widehat{j} + 3\widehat{k})$
D) $\overrightarrow{r} = \widehat{i} + 3\widehat{j} + 2\widehat{k} + \lambda( - \widehat{i} - 5\widehat{j} - 3\widehat{k})$
Explanation

(b)
The required line passes through the point $\widehat{i} + 3\widehat{j} + 2\widehat{k}$ and is perpendicular to the lines $\overrightarrow{r} = \left( \widehat{i} + 2\widehat{j} - \widehat{k} \right) + \lambda(2\widehat{i} + \widehat{j} + \widehat{k})$ and $\overrightarrow{r} = \left( 2\widehat{i} + 6\widehat{j} + \widehat{k} \right) + \mu(\widehat{i} + 2\widehat{j} + 3\widehat{k})$; therefore it is parallel to the vector $\overrightarrow{b} = \left( 2\widehat{i} + 6\widehat{j} + \widehat{k} \right) \times \mu\left( \widehat{i} + 2\widehat{j} + 3\widehat{k} \right) = (\widehat{i} - 5\widehat{j} + 3\widehat{k})$
Hence, the equation of the required line is
$\overrightarrow{r} = \left( \widehat{i} + 3\widehat{j} + 2\widehat{k} \right) + \lambda(\widehat{i} - 5\widehat{j} + 3\widehat{k})$

Dr. Abubacker
Reviewed by Dr. Abubacker

Physics Educator & NEET/JEE Specialist

Ph.D. in Physics with 19+ years of teaching experience.

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