   Chapter 12, Problem 27RE

Chapter
Section
Textbook Problem

# Find the distance between the planes 3x + y − 4z = 2 and 3x + y − 4z = 24.

To determine

To find: The distance between the parallel planes 3x+y4z=2 and 3x+y4z=24.

Explanation

As the planes are parallel, distance between planes is the distance from any point on one plane to the other plane. Therefore, choose any point on one plane and calculate its distance to the other plane.

Write the equation of first plane as follows.

3x+y4z=2 (1)

Consider the values of y- and z-coordinates are 0 and 0 respectively.

Substitute 0 for y and 0 for z in equation (1) and obtain the x-coordinate value.

3x+(0)4(0)=23x+00=23x=2x=23

Therefore, the point (23,0,0) is on the first plane.

Formula used:

Write the expression to find the distance from the point P(x1,y1,z1) to the plane.

D=|ax1+by1+cz1+d|a2+b2+c2 (2)

Here,

x1, y1, and z1 are the coordinates of the point P(x1,y1,z1), which is (23,0,0),

a, b, and c are the normal vector numbers to the plane and

d is the constant parameter.

Write the expression to find constant parameter (d).

ax0+by0+cz0+d=0 (3)

Here,

x0, y0, and z0 are the coordinates of any point P(x0,y0,z0) in the plane

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