Volume of parallelopiped $=\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]$

Here, the parallelopiped is formed by the vectors $\hat{i}+\hat{j}+\hat{k}, \hat{j}+a\hat{k}$ and $a\widehat{i}+\widehat{k}$ 
So, the volume, $V$ $=\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& a\\ a& 0& 1\end{array}\right|=1+a^2-a$

$=\left|\frac{3}{4}+{\left(a-\frac{1}{2}\right)}^2\right|$ cubic units

Hence, the volume of the parallelopiped is minimum when $a=\frac{1}{2}$