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: 27

Answer

(a) The point $(1,-3,4)$ does not lie on the line. (b) The point $(5,5,2)$ lies on the line.

Work Step by Step

$(x,y,z) = (2,-1,5)+n(1,2,-1)$ (a) If $(1,-3,4)$ lies on the line, then there is a real number $n$ such that $(2,-1,5)+n(1,2,-1) = (1,-3,4)$ $x$: If $2+n(1) = 1$, then $n = -1$ $y$: If $-1+n(2) = -3$, then $n = -1$ $z$: If $5+n(-1) = 4$, then $n = 1$ Since the required value of $n$ is not the same for $x,y,$ and $z$, the point $(1,-3,4)$ does not lie on the line. (b) If $(5,5,2)$ lies on the line, then there is a real number $n$ such that $(2,-1,5)+n(1,2,-1) = (5,5,2)$ $x$: If $2+n(1) = 5$, then $n = 3$ $y$: If $-1+n(2) = 5$, then $n = 3$ $z$: If $5+n(-1) = 2$, then $n = 3$ Since the required value of $n$ is the same for $x,y,$ and $z$, the point $(5,5,2)$ lies on the line.
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