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

$\overline{\mathrm{p}}=(1,\mathrm{a},1),\overline{\mathrm{q}}=(0,1,\mathrm{a}),\overline{\mathrm{r}}=(\mathrm{a},0,1)$
$[\overline{p},\overline{q},\overline{r}]=\left|\begin{array}{ccc}1& a& 1\\ 0& 1& a\\ a& 0& 1\end{array}\right|$
$$\begin{gathered} =1\left[1\right]-a[-a^2]+[-a] \\ =a^3-a+1 \\ V=a^3-a+1 \\ f\left(a\right)=a^3-a+1 \\ {f}^{|}\left(a\right)=3a^2-1,f^{\left|\right|}\left(a\right)=6a \\ {f}^{|}\left(a\right)=0\Rightarrow a=\pm \frac{1}{\sqrt{3}} \\ {f}^{\left|\right|}\left(a\right)>0\Rightarrow a=\frac{1}{\sqrt{3}} \end{gathered}$$