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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)

5x +  2y + 3z = 2, y = 4x − 6z

To determine

Whether the planes 5x+2y+3z=2 and y=4x6z 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 the first plane as follows:

5x+2y+3z=2

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

n1=5,2,3

The normal vector n1 is also written as follows:

n1=5i+2j+3k

Write the equation of the second plane as follows:

y=4x6z

Rewrite the expression as follows:

4xy6z=0

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

n2=4,1,6

The normal vector n2 is also written as follows:.

n2=4ij6k

From the normal vectors of the planes, it is clear that the normal vector of one of the plane is not the scalar multiple of other plane.

n1kn2

Therefore, the planes are not parallel.

Find the dot product of two planes (n1n2)

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