Direction Ratios of the first line
$\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& -a& 0\\ 0& c& -1\end{array}\right|=a \widehat{i}+\widehat{j}+c\widehat{k}$
Hence, Direction Ratios of the first line are $a, 1, c.$
For Direction Ratios of the second line
$\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 0& -a^{\prime }\\ 0& 1& -c^{\prime }\end{array}\right|=a^{\prime }\widehat{i}+c^{\prime }\widehat{j}+\widehat{k}$
Hence, Direction Ratios of the $2^{nd}$ line $=a^{\prime }, c^{\prime }, 1$
Since, both the lines are perpendicular,
$a.a^{\prime }+c^{\prime }+c=0$