The value of $a$  so that the volume of the parallelepiped formed by

$\overrightarrow{i}+a\overrightarrow{j}+\overrightarrow{k}$ , $\overrightarrow{j}+a\overrightarrow{k}$ and  $a\overrightarrow{i}+\overrightarrow{k}$  becomes minimum is 

Volume of the parallelepiped is 
$V\left(a\right)=\left|\begin{array}{ccc}1& a& 1\\ 0& 1& a\\ a& 0& 1\end{array}\right|=a^3-a+1$
$V^{\text{'}}\left(a\right)=3a^2-1=0$ if  $a=\pm \frac{1}{\sqrt{3}}$
$V^{"}\left(a\right)=6a>0$ if  $a=\frac{1}{\sqrt{3}}$
Thus $V\left(a\right)$  is minimum when  $a=\frac{1}{\sqrt{3}}.$