Question

If line $\overrightarrow{r}=(\hat{i}-2\hat{j}-\hat{k})+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ is parallel to the plane $\overrightarrow{r}.(3\hat{3}-2\hat{j}-m\hat{k})=14$, then the value of m is

• A. $2$
• B. $-2$
• C. $0$
• D. can not be predicated with these information

Answer 1

The correct option is A $2$

If $\overrightarrow{r}=(\hat{i}-2\hat{j}-\hat{k})+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ is parallel to $\overrightarrow{r}.(3\hat{i}-2\hat{j}-m\hat{k})=14$

Then the plane's Normal Vector should be perpendicular to the line

Hence, their dot product will be 0.

$\Rightarrow (2\hat{i}+\hat{j}+2\hat{k}).(3\hat{i}-2\hat{j}-m\hat{k})=0\Rightarrow 6-2-2m=0\Rightarrow 4=2m\Rightarrow m=2$