## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 12: Vectors and the Geometry of Space - Section 12.1 - Three-Dimensional Coordinate Systems - Exercises 12.1 - Page 696: 31

#### Answer

$a.\quad \left\{\begin{array}{l} y=3,\\ z=-1 \end{array}\right.$ $b.\quad \left\{\begin{array}{l} x=1,\\ z=-1 \end{array}\right.$ $\mathrm{c}.\quad \left\{\begin{array}{l} x=1,\\ y=3 \end{array}\right.$

#### Work Step by Step

$a.\quad$ The x-axis is the intersection of the xz and xy-planes, $\left\{\begin{array}{l} y=0,\\ z=0 \end{array}\right.$ Planes $\left\{\begin{array}{l} y=3,\\ z=-1 \end{array}\right. \quad$ intersect in a line parallel to the x-axis, and the points on this line are $(t,3,-1)$, so the given point is also on this line. $b.\quad$ The y-axis is the intersection of the yz and xy-planes, $\left\{\begin{array}{l} x=0,\\ z=0 \end{array}\right.$ Planes $\left\{\begin{array}{l} x=1,\\ z=-1 \end{array}\right. \quad$ intersect in a line parallel to the x-axis, and the points on this line are $(1,t,-1)$, so the given point is also on this line. $\mathrm{c}.\quad$ The z-axis is the intersection of the yz and xz-planes, $\left\{\begin{array}{l} x=0,\\ y=0 \end{array}\right.$ Planes $\left\{\begin{array}{l} x=1,\\ y=3 \end{array}\right. \quad$ intersect in a line parallel to the x-axis, and the points on this line are $(1,3,t)$, so the given point is also on this line.

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