If 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 perpendicular to each then the value of a  can be :

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

$\left(a\hat{i}+a\hat{j}+3\hat{k}\right).\left(a\hat{i}-2\hat{j}-\hat{k}\right)=0$

$a.a-a.2-3.1=0$

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

$a^2-3a+a-3=0$

$\left(a-3\right)\left(a+1\right)=0$

$a=3$  and $a=-1$