$\text{ The angle between the lines }x-3y-4=0,4y-z+5=0\text{ and }x+3y-11=0,2y-z+6=0\text{ is }$

$\text{ Line }x-3y-4=0=4y-z+5\text{ is line of intersection of planes }x-3y-4=0\text{ and }4y-z+5=0$

Vector along line of intersection is
$\left|\begin{array}{r}\widehat{i}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{j}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\widehat{k}\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\end{array}\right|=3\widehat{i}+\widehat{j}+4\widehat{k}=\overrightarrow{a}$

$\text{ Line }x+3y-11=0=2y-z+6\text{ is line of intersection of planes }x+3y-11=0\text{ and }2y-z+6=0$

Vector along line of intersection is
$\begin{array}{r}\left|\begin{array}{lll}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 3& 0\\ 0& 2& -1\end{array}\right|=-3\widehat{i}+\widehat{j}+2\widehat{k}=\overrightarrow{b}\\ \overrightarrow{a}\cdot \overrightarrow{b}=\left(3\right)(-3)+\left(1\right)\left(1\right)+\left(4\right)\left(2\right)=0\end{array}$