Question

If vectors $\overrightarrow{P}=a\hat{i}+a\hat{j}+3\hat{k}\text{and}\overrightarrow{Q}=a\hat{i}-2\hat{j}-\hat{k}$ are perpendicular to each other, then the positive value of $a$ is

• A. $\text{Zero}$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

If $\overrightarrow{P}$ and $\overrightarrow{Q}$ are perpendicular to each other then $\overrightarrow{P}.\overrightarrow{Q}=0$

Given:
$\overrightarrow{P}=a\hat{i}+a\hat{j}+3\hat{k}$
$\overrightarrow{Q}=a\hat{i}-2\hat{j}-\hat{k}$
$\therefore (a\hat{i}+a\hat{j}+3\hat{k})(a\hat{i}-2\hat{j}-\hat{k})=0$
$\Rightarrow a^2-2a-3=0\Rightarrow (a-3)(a+1)=0$
$\Rightarrow a=3$ and $a=-1$