Question

If the planes $ \overrightarrow{r}.(2\hat{i}-\lambda \hat{j}+3\hat{k})=0 $ and $ \overrightarrow{r}.(\lambda \hat{i}+5\hat{j}-\hat{k})=0 $ are perpendicular to each other, then the value of $ {{\lambda }^{2}}+\lambda $ is

• A. 0
• B. 2
• C. 1
• D. 3
• E. 4

Answer 1

Answer: (A)

Since, given planes $ \overrightarrow{r}.(2\hat{i}-\lambda \hat{j}+3\hat{k})=0 $ and $ \overrightarrow{r}.(\lambda \hat{i}+5\hat{j}-\hat{k})=5 $
are perpendicular.
$ \therefore $ $ 2(\lambda )-\lambda (5)+3(-1)=0 $
$ \Rightarrow $ $ -3\lambda -3=0\Rightarrow \lambda =-1 $
$ \therefore $ $ {{\lambda }^{2}}+\lambda ={{(-1)}^{2}}-1 $
$=0 $