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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find symmetric equations for the line of intersection of the planes.

z = 2xy − 5, z = 4x + 3y − 5

To determine

To find: The symmetric equations for the line of intersection of the planes z=2xy5 and z=4x+3y5.

Explanation

Formula used:

The symmetric equations of a line through the point (x0,y0,z0) and parallel to the direction vector a,b,c is,

xx0a=yy0b=zz0c (1)

Calculation:

Write the equation of first plane as follows.

2xyz=5 (2)

Write the equation of second plane as follows.

4x+3yz=5 (3)

Consider a point on the line of intersection P0(x0,y0,z0) such that the point satisfies the equations of both the planes z=2xy5 and z=4x+3y5.

Set z=0 and solve the two equations of the intersecting planes.

Substitute 0 for z in equation (2),

2xy(0)=52xy=5

y=2x+5 (4)

Substitute 0 for z in equation (3),

4x+3y(0)=5

4x+3y=5 (5)

Substitute equation (4) in (5) and obtain the x- and y-coordinates as follows.

x=2y=1

Therefore, the point on the line of intersection P0(x0,y0,z0) is (2,1,0).

The parallel direction vector is the cross product of normal vectors of both the planes.

v=n1×n2

The expression is also written as follows.

v=|ijka1b1c1a2b2c2| (6)

The normal vector (n1) of first plane.

n1=2ijk

The normal vector of first plane is also written as follows.

n1=2,1,1

The normal vector (n2) of second plane.

n2=4i+3jk

The normal vector of second plane is also written as follows

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Chapter 12 Solutions

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P-48ESect-12.6 P-49ESect-12.6 P-50ESect-12.6 P-51ESect-12.6 P-52ECh-12 P-1RCCCh-12 P-2RCCCh-12 P-3RCCCh-12 P-4RCCCh-12 P-5RCCCh-12 P-6RCCCh-12 P-7RCCCh-12 P-8RCCCh-12 P-9RCCCh-12 P-10RCCCh-12 P-11RCCCh-12 P-12RCCCh-12 P-13RCCCh-12 P-14RCCCh-12 P-15RCCCh-12 P-16RCCCh-12 P-17RCCCh-12 P-18RCCCh-12 P-19RCCCh-12 P-1RQCh-12 P-2RQCh-12 P-3RQCh-12 P-4RQCh-12 P-5RQCh-12 P-6RQCh-12 P-7RQCh-12 P-8RQCh-12 P-9RQCh-12 P-10RQCh-12 P-11RQCh-12 P-12RQCh-12 P-13RQCh-12 P-14RQCh-12 P-15RQCh-12 P-16RQCh-12 P-17RQCh-12 P-18RQCh-12 P-19RQCh-12 P-20RQCh-12 P-21RQCh-12 P-22RQCh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-1PCh-12 P-2PCh-12 P-5P

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