Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.5 Equations of Lines and Planes - 12.5 Exercises - Page 873: 82

Answer

(a) $x+y+z=c$ This is the family of the planes that have normal vectors parallel to $ \lt 1,1,1\gt$. The distance from the plane to the origin varies by changing the value of $c$. (b) $x+y+cz=1$ This is the family of the planes that intercepts the $xy$-plane at the 2D line $x+y=1$. The distance from the plane to the By varying the value of $c$ makes the plane rotate around the line. (c) $y cos \theta+zsin \theta=1$ Because there is no $x$, the planes will generates will be parallel to $x-$ axis. These lines are basically tangent lines to a circle of radius $1$ So, the planes are the tangent lines to a circle of radius $1$ on $yz$ plane extended in the directions parallel to the $x-axis$.

Work Step by Step

(a) $x+y+z=c$ This is the family of the planes that have normal vectors parallel to $ \lt 1,1,1\gt$. The distance from the plane to the origin varies by changing the value of $c$. (b) $x+y+cz=1$ This is the family of the planes that intercepts the $xy$-plane at the 2D line $x+y=1$. The distance from the plane to the By varying the value of $c$ makes the plane rotate around the line. (c) $y cos \theta+zsin \theta=1$ Because there is no $x$, the planes will generates will be parallel to $x-$ axis. These lines are basically tangent lines to a circle of radius $1$ So, the planes are the tangent lines to a circle of radius $1$ on $yz$ plane extended in the directions parallel to the $x-axis$.
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