   Chapter 12.5, Problem 49E

Chapter
Section
Textbook Problem

# Find direction numbers for the line of intersection of the planes x + y + z = 1 and x + z = 0.

To determine

To find: The direction numbers for the line of intersection of the planes x+y+z=1 and x+z=0 .

Explanation

The direction vector of the line is the cross product of normal vectors of the intersected planes.

Formula:

Write the expression to find the direction vector (v) for the line of intersection of the planes.

v=n1×n2 (1)

Here,

n1 is the normal vector of the first plane and

n2 is the normal vector of the second plane.

Write the equation of the first plane as follows:

x+y+z=1

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

n1=1,1,1

Write the equation of the second plane as follows:

x+z=0

Rewrite the expression as follows:

x+(0)y+z=0

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

n2=1,0,1

Calculation of direction vector of line (v) :

Substitute 1,1,1 for n1 and 1,0,1 for n2 in equation (1),

v=1,1,1×1,0,1

The cross product of normal vectors is solved as follows

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