If $l, m, n$ are the direction cosines of a line that is perpendicular to the lines having the direction…
If $l, m, n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $1,2,-1$ and $1,-2,1$ then $(l+m+n)^2=$
$\frac{1}{20}$
$\frac{9}{5}$
$\frac{1}{5}$
$\frac{3}{20}$
Solution
$(l, m, n)$ are direction cosine of a line that is perpendicular to the lines having direction ratios $(1,2,-1)$ and $(1,-2,1)$
$\therefore l+2 m-n=0$ ...(i)
$l-2 m+n=0$ ...(ii)
Adding (i) and (ii), $2 l=0 \Rightarrow l=0$
From (i), $2 m-n=0 \Rightarrow 2 m=n$
Also, $l^2+m^2+n^2=1 \Rightarrow m^2+4 m^2=1 \Rightarrow m^2=\frac{1}{5}$
Now, $(l+m+n)^2=l^2+m^2+n^2+2(l m+m n+l n)$
$=1+2(m n)=1+4 m^2=\frac{9}{5}$