The distance between two planes is given by the length of the normal vector falling from one plane to the other plane. We can also find the distance between two planes using the formula for the distance between a point and a plane, by taking a point in one plane and taking the distance from the other plane. The formula for the distance between two parallel planes π_{1}: ax + poke + cz + d_{1}= 0 e p_{2}: ax + poke + cz + d_{2}= 0 is |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}).

Let's learn how to determine the distance between two planes formula and the distance between two parallel planes using the point-plane distance formula. We will also learn how to apply the formulas using some examples for a better understanding of the concept.

1. | What is the distance between two planes? |

2. | Distance formula between two planes |

3. | Distance between two faces using the point-to-face distance formula |

4. | Apply distance formulas between two planes |

5. | Frequently asked questions about the distance between two planes |

## What is the distance between two planes?

The distance between twoairplanescan be determined by the smallest distance between the surfaces of the two planes. We can have two parallel planes or non-parallel planes. Since the distance between two planes is the shortest distance between them, planes that are not parallel intersect and hence their distance is zero. To find the distance between two parallel planes, we can calculate the length of the perpendicularvectorbetween the surfaces of the two planes. Now let's look at the formula to determine the distance between two planes.

## Distance formula between two planes

The distance between two planes is the length of the normal vector between them. Now we have two types of planes: parallel planes and non-parallel planes. So, to determine the distance between two planes, we go through the formulas for determining the distance between two parallel planes and two non-parallel planes.

### Distance between two parallel planes

The formula for the distance between two parallel planes is similar to the formula for determining the distance between two parallel lines. As we know, the coordinates of the normal vectors of the two parallel planes are proportional or equal. So consider the equations of two parallel planes as P_{1}: ax + poke + cz + d_{1}= 0 in P_{2}: ax + poke + cz + d_{2}= 0. The formula for the distance between two parallel planes is thus given by: |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}). Note that if the coefficients a, b, c are not equal, we make them equal using the common ratio a_{1}/A_{2}= geb_{1}/B_{2}= c_{1}/C_{2}to obtain the equivalent equation of the plane.

### Distance between two non-parallel planes

As we know, the distance between two planes is nothing more than the shortest distance between them. Therefore, if two planes are not parallel, they must intersect in a straight line in three-dimensional space. Therefore, the shortest distance between two non-parallel planes is zero. Therefore, the distance between two planes that are not parallel is always zero.

## Distance between two faces using the point-to-face distance formula

Next, we examine the method of determining the distance between the two planes using the point-plane distance formula. The formula for calculating the distance between a point (x_{1}, You_{1}, z_{1}) to a plane ax + by + cz + d = 0 is given by D = |ax_{1}+door_{1}+cz_{1}+d |√(a^{2}+b^{2}+c^{2}). To find the distance between two faces using the point-to-face distance formula, we can follow the steps below:

- Step 1: Convert the equations of the two planes to the standard format i.e. ax + by + cz + d = 0
- Step 2: Check that the planes are parallel. [Two P-planes
_{1}: A_{1}x + born_{1}a + c_{1}z + d_{1}= 0 in P_{2}: A_{2}x + born_{2}a + c_{2}z + d_{2}= 0 are parallel to a_{1}/A_{2}= geb_{1}/B_{2}= c_{1}/C_{2}] - Step 3: Consider the coefficients a, b, c, d of the equation of one of the planes.
- Step 4: Consider a point P(x
_{1}, You_{1}, z_{1}) on the other plane. [An easy way to find the point is to take x = y = 0 and find the value of z from thecomparison of the other plane] - Step 5: Replace the values of a, b, c, d, x
_{1}, You_{1}, z_{1}in the formula for the distance between a point and a plane: |ax_{1}+door_{1}+cz_{1}+d |√(a^{2}+b^{2}+c^{2})

By following the above steps, we can find the distance between two planes using the formula fordistance between point and plane.

## Apply distance formulas between two planes

Now that we know the two methods to find the distance between two planes, let's solve some examples based on these methods to understand their application.

**Example 1:**Calculate the distance between two planes P_{1}: 2x + 4y + z + 7 = 0 door P_{2}: 4x + 8j + 2z - 14 = 0.

**Solution:**First, let's see if the planes are parallel. Take the ratio of the coefficients in the equations of the two planes. Here we have one_{1}= 2, geb_{1}= 4, ca_{1}= 1 and one_{2}= 4,b_{2}= 8, ca_{2}= 2. Therefore, we have, the_{1}/A_{2}= geb_{1}/B_{2}= c_{1}/C_{2}= 1/2 ⇒ The two planes are parallel. Now to get equal coefficients in the equations of the two planes, divide the equation of P_{2}by 2. So we have P_{2}: (1/2)(4x + 8y + 2z - 14) = (1/2) (0) ⇒ P_{2}: 2x + 4y + z - 7 = 0. Now we have a = 2, b = 4, c = 1, d_{1}= 7,d_{2}= -7. Now the formula to find the distance between two planes P_{1}e P_{2}is: |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}). Therefore, the required distance is,

d = |-7 - 7|/√(2^{2}+4^{2}+1^{2})

= |-14|/√(4 + 16 + 1)

= 14/√(21)

= (2/3)√21 units

Next, let's find the distance between two faces using the point-to-face distance formula.

**Example 2:**Calculate the distance between two planes P_{1}: 2x + 4y + z + 7 = 0 door P_{2}: 4x + 8j + 2z - 14 = 0.

**Solution:**As we verified in Example 1, the two planes P_{1}e P_{2}are parallel. Consider a = 2, b = 4, c = 1, d = 7 from the foreground. Then we find a point (x_{1}, You_{1}, z_{1}) on the other plane. For this, assume that x_{1}= j_{1}= 0, and plug these values into the equation for P_{2}, We have,

4(0) + 8(0) + 2z_{1}- 14 = 0

⇒ 2z_{1}- 14 = 0

⇒ z_{1}= 7

So we have a point (0, 0, 7) in the plane P_{2}and the equation of the foreground P_{1}: 2x + 4y + z + 7 = 0. Now let's find the distance between the point (0, 0, 7) and the plane P_{1}: 2x + 4y + z + 7 = 0 with the formula |ax_{1}+door_{1}+cz_{1}+d |√(a^{2}+b^{2}+c^{2}).

d = |ax_{1}+door_{1}+cz_{1}+d |√(a^{2}+b^{2}+c^{2})

= |2 × 0 + 4 × 0 + 1 × 7 + 7|/√(2^{2}+4^{2}+1^{2})

= 14/√21

= (2/3)√21 units

So we have the same distance between the two planes P_{1}e P_{2}using both methods, i.e. (2/3)√21 units

**Important notes Distance between two planes**

- The distance between two planes is zero when they intersect.
- The distance between two planes P
_{1}: ax + poke + cz + d_{1}= 0 in P_{2}: ax + poke + cz + d_{2}= 0 that are parallel is given by: |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}) - The distance between two parallel planes can also be calculated using the point-to-plane distance formula.

**Topics related to the distance between two planes**

- Distance between two points
- Euclidean distance formula
- Geometry

## Frequently asked questions about the distance between two planes

### What is the distance between two planes in geometry?

The distance between two planes is given by the length of the normal vector falling from one plane to the other plane and can be determined by the smallest distance between the surfaces of the two planes.

### How to find the distance between two planes?

The distance between two planes can be determined in two ways. We can use the |d formula_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}) or using the point-to-face spacing formula.

### What is the formula for the distance between two planes?

Consider the equations of two parallel planes as P_{1}: ax + poke + cz + d_{1}= 0 in P_{2}: ax + poke + cz + d_{2}= 0. The formula for the distance between two parallel planes is thus given by d = |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2}).

### What is the distance between two parallel planes?

The distance between two parallel planes P_{1}: ax + poke + cz + d_{1}= 0 in P_{2}: ax + poke + cz + d_{2}= 0 can be determined with the formula d = |d_{2}-D_{1}|/√(one^{2}+b^{2}+c^{2})

### How do you find the distance between two non-parallel planes?

The distance between two planes is nothing more than the shortest distance between them. Therefore, if two planes are not parallel, they must intersect. Therefore, the distance between two planes that are not parallel is always zero.

## FAQs

### Distance between two planes - formula, examples | Distance between parallel planes? ›

The formula for determining the distance between two planes π1: ax + by + cz + d1 = 0 and π2: ax + by + cz + d2 = 0 is **|d2 – d1|/√(a2 + b2+ c2)**. The shortest distance between the surfaces of two planes can be used to calculate the distance between them. Two parallel planes or two non-parallel planes are possible.

**What is the formula for distance between two parallel lines? ›**

The formula for distance between two parallel lines having the slope-intercept form of equations of the two lines as y = mx + c_{1} and y = mx + c_{2}, is **d=|c2−c1|√1+m2** d = | c 2 − c 1 | 1 + m 2 .

**What is the equation for parallel planes? ›**

Two planes are parallel if their normal vectors are parallel (constant multiples of one another). It is easy to recognize parallel planes written in the form **ax+by+cz=d** since a quick comparison of the normal vectors n=<a,b,c> can be made.

**What is an example of 2 parallel planes? ›**

**The opposite walls of a room**, floor and ceiling are the examples of parallel planes.

**What is the distance between two adjacent parallel planes? ›**

(2) Now we prove that the distance between two adjacent parallel planes of the direct lattice is **d=2π/G**.

**What is the distance between two parallel planes Crystal? ›**

These are sets of parallel planes within a crystal. The distance between adjacent lattice planes is the **d-spacing**. Note that this can be simplified if a=b (tetragonal symmetry) or a=b=c (cubic symmetry). Example: A cubic crystal has a = 5.2Ε.

**Do parallel planes have the same equation? ›**

determines a point on the line. Consequently, in vector form the equation of the line is r(t)=tv+b=t⟨1,4,−2⟩+⟨2,−4,3⟩. Example 3: Find an equation for the plane passing through the point Q(1,1,1) and parallel to the plane 2x+3y+z = 5. Solution: **parallel planes have the same normal**.

**When two planes are parallel? ›**

Two distinct planes are parallel **if they have parallel nonzero normal vectors**, which means that they have no points of intersection. Two planes are perpendicular if their normal vectors are perpendicular.

**Are two planes parallel to a line parallel? ›**

**Lines and planes are parallel to one another** as in the ordinary geometry: two lines when they lie in one plane and do not intersect, a line and a plane or two planes when they lie in one hyperplane and do not intersect. THEOREM 1. Two lines perpendicular to the same hyperplane are parallel.

**How do you write a plane equation? ›**

The general form of the equation of a plane in ℝ is **𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0** , where 𝑎 , 𝑏 , and 𝑐 are the components of the normal vector ⃑ 𝑛 = ( 𝑎 , 𝑏 , 𝑐 ) , which is perpendicular to the plane or any vector parallel to the plane.

### What is the equation of a plane parallel to the XY plane? ›

**x = a** ; where a is constant. No worries!

**What is parallel plane motion? ›**

A planar motion, is **a motion in which all points of the body move in planes parallel to a certain plane called the plane of the planar (directing) motion**.