Question

If $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$, $\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-({\lambda }^2+3\lambda )\hat{k}$ (where $\lambda$ is constant) and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}-\lambda \overrightarrow{b}$, then sum of different values of $\lambda$ is -

• A. $-1$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $-1$

$(2\hat{i}-\hat{j}+\hat{k}).\left[\right(1-\lambda )\hat{i}+(3-\lambda )\hat{j}+(2\lambda -{\lambda }^2-3\lambda \left)\hat{k}\right]=0$

$\Rightarrow 2-2\lambda -3+\lambda +2\lambda -{\lambda }^2-3\lambda =0$

$\Rightarrow -{\lambda }^2-2\lambda -1=0\Rightarrow (\lambda +1{)}^2=0$

$\Rightarrow \lambda =-1$