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.
$\therefore$ $2(\lambda )-\lambda (5)+3(-1)=0$
$\Rightarrow$ $-3\lambda -3=0\Rightarrow \lambda =-1$
$\therefore$ ${\lambda }^2+\lambda ={(-1)}^2-1$
$=0$