The volume of parallelopiped formed by vectors $\overline{i}+a\overline{j}+\overline{k}$,  $\overline{j}+a\overline{k}, a\overline{i}+\overline{k}$ becomes minimum then a =

$\overline{p}=(1,a,1),\overline{q}=(0,1,a),\overline{r}=(a,0,1)$

$[\overline{p},\overline{q},\overline{r}]=\left|\begin{array}{ccc}1& a& 1\\ 0& 1& a\\ a& 0& 1\end{array}\right|$

$=1\left[1\right]-a[-a^2]+[-a]$

$=a^3-a+1$

$V=a^3-a+1$

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

$f^{|}\left(a\right)=3a^2-1,f^{\left|\right|}\left(a\right)=6a$

$f^{|}\left(a\right)=0\Rightarrow a=\pm \frac{1}{\sqrt{3}}$

$f^{\left|\right|}\left(a\right)>0\Rightarrow a=\frac{1}{\sqrt{3}}$