Answer
The two lines are perpendicular.
Work Step by Step
$l_1: (x,y,z) = (1,-2,5)+n(4,1,-3)$
The direction vector is $(a,b,c) = (4,1,-3)$
$l_2: (x,y,z) = (1,-2,5)+r(-3,6,-2)$
The direction vector is $(d,e,f) = (-3,6,-2)$
We can verify the condition stated in the theorem:
$ad+be+cf = (4)(-3)+(1)(6)+(-3)(-2)$
$ad+be+cf = (-12)+(6)+(6)$
$ad+be+cf = 0$
Since the direction vectors of the two lines satisfy the condition stated in the theorem, the two lines are perpendicular.
