Answer
The two lines are perpendicular.
Work Step by Step
$l_1: (x,y,z) = (2,3,4)+n(1,1,2)$
The direction vector is $(a,b,c) = (1,1,2)$
$l_2: (x,y,z) = (2,3,4)+r(-2,-4,3)$
The direction vector is $(d,e,f) = (-2,-4,3)$
We can verify the condition stated in the theorem:
$ad+be+cf = (1)(-2)+(1)(-4)+(2)(3)$
$ad+be+cf = (-2)+(-4)+(6)$
$ad+be+cf = 0$
Since the direction vectors of the two lines satisfy the condition stated in the theorem, the two lines are perpendicular.
