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# Show that the straight lines whose direction cosines are given by $2l+2m-n=0\;and\;mn+nl+lm=0$ are at right angles.

Toolbox:
• Solve the two equations to get the d.r of both the lines $d_1, d_2$
• Prove $d_1.d_2=0$ for right angle.
Step 1
$2l+2m=n$ ...............(1) put n in
$mn+nl+lm=0$......(2)
Step 2
$2lm+2m^2+2l^2+2lm+lm=0$
Step 3
$2m^2+5lm+2l^2=0$
Step 4
factorise and get
$(m+2l)(2m+l)=0$
$\Rightarrow m= -2l\: or \: m=\large\frac{-l}{2}$
Step 5
Put $l=1\: then \: d_1=(1, -2, -2)\: and \: d_2 ( 1, \large\frac{-1}{2}, 1)$
Step 6
$d_1.d_2=0$

edited Apr 5, 2013