Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 10 - Section 10.6 - The Three-Dimensional Coordinate System - Exercises - Page 482: 31

Answer

Since the lines share at least one point and the lines are parallel, the lines are coincident.

Work Step by Step

When $n=1$, line 1 passes through the point $(1,2,-3)$. Line 2 also passes through this point when $r=0$. Therefore, the lines have at least one point in common. A direction vector for line 1 is $(1,2,-3)$ and a direction vector for line 2 is $(-1,-2,3)$. When $n=-1$, the product of $n$ and the direction vector of line 1 is equal to the direction vector of line 2. Therefore, the lines are parallel. Since the lines share at least one point and the lines are parallel, the lines are coincident.
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