How to Find the Distance between Two Planes

How to Find the Distance between Two Planes
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  • 0:04 Finding Distance…
  • 2:41 The Final Result
  • 3:21 Applying The Result
  • 5:48 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we show how to find the distance between two planes. You will see how to determine the parameters for substituting into the distance formula.

Finding the Distance Between Two Planes

Think about a wall. A wall is flat surface. If the wall extended in all directions to infinity, we would have a plane. A wall usually has a vertical orientation, but a plane can have any orientation. The distance between two planes is the shortest distance between the surfaces of the planes.

Let's find this distance!

Step 1: Write the equations for each plane in the standard format.

The standard format we will use is ax + by+cz + d = 0. For two equations, we have the following:


standard_form


Step 2: Determine if the planes are parallel.

If the planes are not parallel, they will eventually cross because each plane extends to infinity in all directions. Thus, if the planes aren't parallel, the distance between the planes is zero and we can stop the distance finding process. To find out if the planes are parallel, check the ratios a1/a2, b1/b2 and c1/c2.

The planes are parallel if these ratios are equal.


parallel_planes


If the planes are parallel, we continue with the next step.

Step 3: From one of the plane equations, identify the coefficients a, b, c and d.

This part is real easy. Let's say we selected the first plane equation. Then,

  • a is a1
  • b is b1
  • c is c1

Step 4: Find a point (x1, y1, z1) in the other plane.

Any point will do. A good choice is usually to let any of the two coordinates be zero and then solve for the third coordinate. For example, letting x = y = 0 means the following:


a_point_in_the_other_plane


and solving for z1 means:


a_point_in_the_other_plane


Thus,

  • x1 = 0
  • y1 = 0
  • z1 = -d2/c2

Note: This part may look complicated because of all the letter variables. In practice, we have numbers and the work looks a lot simpler. We will show this later in the application section.

Step 5: Substitute into the distance formula and simplify.

Substitute for a, b, c, d, x1, y1 and z1 into the formula for finding the distance, D, between two planes.

The Final Result

As you can see now, we have the formula for distance:


the_distance_formula


In this formula, a, b, c and d are the coefficients of the equation describing one of the planes and x1, y1 and z1 are the coordinates of a point in the other plane.

The format for the equation of the plane is ax + by + cz + d = 0.

If the planes are not parallel, the distance is zero.

Applying the Result

Example 1: Find the distance between the two planes: 2x + 4y + 6z + 8 = 0 and 4x + 8y + 2z - 16 = 0.

Both equations are already in the standard format. We now check the ratios of coefficients:

  • ratio of x coefficients: 2/4 = 1/2
  • ratio of y coefficients: 4/8 = 1/2
  • ratio of z coefficients: 6/2 = 3

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