# Chapter 10 - Section 10.6 - The Three-Dimensional Coordinate System - Exercises - Page 482: 33

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.

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