Show that the three lines with direction cosines

$\cfrac{{12}}{{13}},\cfrac{{ - 3}}{{13}},\cfrac{{ - 4}}{{13}};\cfrac{4}{{13}},\cfrac{{12}}{{13}},\cfrac{3}{{13}};\cfrac{3}{{13}},\cfrac{{ - 4}}{{13}},\cfrac{{12}}{{13}}$ are mutually perpendicular.

Let the lines whose direction cosines are given be ${l_1},{l_2}$ and ${l_3}.$

Let $\alpha$be the angle between ${l_1}$ and ${l_2},$

then
$\cos \alpha = \left| {\left( {\cfrac{{12}}{{13}}} \right) \cdot \left( {\cfrac{4}{{13}}} \right) + \left( {\cfrac{{ - 3}}{{13}}} \right)\left( {\cfrac{{12}}{{13}}} \right) + \left( {\cfrac{{ - 4}}{{13}}} \right) \cdot \left( {\cfrac{3}{{13}}} \right)} \right|$

$= \cfrac{{48 - 36 - 12}}{{169}} = 0 \Rightarrow$ $\alpha = \cfrac{\pi }{2} \Rightarrow {l_1} \bot {l_2}$

Similarly, we can show that ${l_2} \bot {l_3}$ and ${l_3} \bot {l_1}$