Question

The vectors, $\overrightarrow{P}=a\hat{i}+a\hat{j}+3\hat{k}$ and $\overrightarrow{Q}=a\hat{i}-2\hat{j}-\hat{k}$ are mutually perpendicular. The value of $a$ will be -

• A. $3$
• B. $4$
• C. $9$
• D. $1$

Answer 1

The correct option is A $3$

Given:

$\overrightarrow{P}=a\hat{i}+a\hat{j}+3\hat{k}$

$\overrightarrow{Q}=a\hat{i}-2\hat{j}-\hat{k}$

As $\overrightarrow{P}\perp \overrightarrow{Q}\Rightarrow \overrightarrow{P}\cdot \overrightarrow{Q}=0$

$\therefore (a\times a)+(a\times -2)+(3\times -1)=0$

$\Rightarrow a^2-2a-3=0$

$\Rightarrow (a-3)(a+1)=0$

$\Rightarrow a=3$ or $a=-1$

Hence, option $\left(A\right)$ is the correct answer.