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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes.

x + y + z = 1, x + 2y + 2z = l

(a)

To determine

To find: The parametric equations for the line of intersection of the planes x+y+z=1 and x+2y+2z=1 .

Explanation

Formula:

Write the expressions to find the parametric equations for a line through the point (x0,y0,z0) and parallel to the direction vector a,b,c .

x=x0+at,y=y0+bt,z=z0+ct (1)

Write the equation of first plane as follows.

x+y+z=1 (2)

Write the equation of second plane as follows.

x+2y+2z=1 (3)

Consider a point on the line of intersection P0(x0,y0,z0) such that the point satisfies the equations of both the planes x+y+z=1 and x+2y+2z=1 .

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

Substitute 0 for z in equation (2),

x+y+(0)=1

x+y=1 (4)

Calculation of point on the line of intersection of planes:

Substitute 0 for z in equation (3),

x+2y+2(0)=1

x+2y=1 (5)

Solve equations (4) and (5) and obtain the x- and y-coordinates as follows.

x=1y=0

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

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)

Write the normal vector (n1) from the equation (2) of first plane.

n1=i+j+k

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

n1=1,1,1

Write the normal vector (n2) from the equation (3) of second plane

(b)

To determine

To find: The angle between the planes x+y+z=1 and x+2y+2z=1 .

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Chapter 12 Solutions

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