I have the following question before me: Let $l$, $m$ &$n$ be the direction cosines of two lines given by equations $l+m+n=0$. $fmn+gln+hlm=0$. I have to prove that for the angle between the two lines to be $π/3$, the following condition is satisfied: $1/f +1/g +1/h=0$.
I substituted the value of $l$ as $l=-(m+n)$ in the second equation and got a quadratic equation in $m/n$ by dividing the resulting equation by $n^2$. Let $l_1$ , $m_1$ and $n_1$ be the direction cosines of first line and $l_2$, $m_2$ and $n_2$ be those of second line.
From the quadratic equation, I get $m_1 m_2/n_1 n_2= g/h$. From symmetry of the equations, I can write $m_1 m_2/l_1 l_2= g/f$. Also we can write the angle condition as : $l_1 l_2+ m_1 m_2+ n_1 n_2= cos(π/3)=1/2$.
But I do not know how to proceed from here to arrive at the required condition. Please suggest.