Question

Find the value of $m$ for which the line $\overrightarrow{r}=(\hat{i}+2\hat{k})+\lambda (2\hat{i}-m\hat{j}-3\hat{k})$ is parallel to the plane $\overrightarrow{r}\cdot (m\hat{i}+3\hat{j}+\hat{k})=4$ ?

• (A) 2
• (B) -2
• (C) 3
• (D) -3

Answer 1

The Correct Option is D

Explanation:
Given: Equation of line $\overrightarrow{r}=(\hat{i}+2\hat{k})+\lambda (2\hat{i}-m\hat{j}-3\hat{k})$ and equation of plane $\overrightarrow{r}\cdot (m\hat{i}+3\hat{j}+\hat{k})=4$It is given that the given line is parallel to the given plane,As we know that, if the line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ is parallel to the plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$ then $\overrightarrow{b}\cdot \overrightarrow{n}=0$Here, $\overrightarrow{b}=2\hat{i}-m\hat{j}-3\hat{k}$ and $\overrightarrow{n}=m\hat{i}+3\hat{j}+\hat{k}$ $\Rightarrow \overrightarrow{b}\cdot \overrightarrow{n}=2m-3m-3=0$ $\Rightarrow m=-3$Hence, the correct option is (D).