   Chapter 12, Problem 24RE

Chapter
Section
Textbook Problem

# (a) Show that the planes x + y − z = 1 and 2x − 3y + 4z = 5 are neither parallel nor perpendicular.(b) Find, correct to the nearest degree, the angle between these planes.

(a)

To determine

To show: The planes x+yz=1 and 2x3y+4z=5 are neither parallel nor perpendicular.

Explanation

If the two planes are parallel, the normal vector of one of the planes is the scalar multiple of the 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:

x+yz=1

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

n1=1,1,1

The normal vector n1 is also written as follows:

n1=i+jk

Write the equation of second plane as follows:

2x3y+4z=5

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

n2=2,3,4

The normal vector n2 is also written as follows:

n2=2i3j+4k

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

n1kn2

Therefore, the planes are not parallel.

Find the dot product of the two planes (n1n2)

(b)

To determine

To find: The angle between the planes x+yz=1 and 2x3y+4z=5.

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