   Chapter 12.5, Problem 55E

Chapter
Section
Textbook Problem

# Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)2x − 3y = z, 4x = 3 + 6y + 2z

To determine

Whether the planes 2x3y=z and 4x=3+6y+2z are parallel, perpendicular, or neither.

Explanation

If the two planes are parallel, the normal vector of one of the planes is the scalar multiple of normal vector of the other plane.

If the two planes are perpendicular to each other, the dot product of the normal vectors of the two planes is zero (the normal vectors of the planes must be orthogonal).

Write the equation of first plane as follows:

2x3y=z

Rewrite the expression as follows:

2x3yz=0

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

n1=2,3,1

The normal vector n1 is also written as follows:

n1=2i3jk

Write the equation of second plane as follows:

4x=3+6y+2z

Rewrite the expression as follows:

4x6y2z=3

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

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