# The angle between the line formed by $3x-7y-5z=1$ and $8x-11y+2z=0$ and the line formed by $5x-13y+3z+2=0$ and $8x-11y+2z=0$

$\large(a)\:\:\frac{\pi}{3}\:\:\:\qquad\:\:(b)\:\:\frac{\pi}{4}\:\:\:\qquad\:\:(c)\:\:\frac{\pi}{2}\:\:\:\qquad\:\:(d)\:\:\pi$

Given equations of the planes be
$3x-7y-5z-1=0..........(i)$
$8x-11y+2z=0........(ii)$
$5x-13y+3z+2=0..........(iii)$
Let the $d.r.$ of the line formed by $(i)$ and $(ii)$, $L_1$ be $(l_1,m_2,n_2)$ and
let the $d.r.$ of the line formed by $(ii)$ and $(iii)$, $L_2$ be $(l_2,m_2,n_2)$
$\Rightarrow\:$ $(l_1,m_1,n_1)\:\:is\:\:\perp$ to $(3,-7,-5)$ and $(8,-11,2)$
$\Rightarrow\: (l_1,m_1,n_1)=(3,-7,-5)\times(8,-11,2)=(-69,-46,23)=(-3,-2,1)$
Similarly $(l_2,m_2,n_2)\:\:is\:\perp$ to $(5,-13,3)\:\:and\:\:(8,-11,2)$
$\Rightarrow\:(l_2,m_2,n_2)=(5,-13,3)\times(8,-11,2)=(7,14,49)=(1,2,7)$
$\therefore$ The angle between $L_1$ and $L_2$ is given by
$cos\theta=\large\frac{(l_1,m_1,n_!).(l_2,m_2,n_2)}{|(l_1,m_1,n_1)||(l_2,m_2,n_2)|}$
$=\large\frac{(-3,-2,1).(1,2,7)}{\sqrt {14}.\sqrt { 54}}$$=0$
$\therefore$ $\theta=\large\frac{\pi}{2}$