Distance Between Two Planes

The distance between two planes is given by the length of the normal vector that drops from one plane onto the other plane. We can also find the distance between two planes using the formula for the distance between a point and plane by considering a point on one plane and taking its distance from the other plane. The formula for the distance between two parallel planes π₁: ax + by + cz + d₁ = 0 and π₂: ax + by + cz + d₂ = 0 is |d₂ - d₁|/√(a² + b² + c²).

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

The distance between two planes can be determined by the shortest distance between the surfaces of the two planes. We can either have two parallel planes or non-parallel planes. Since the distance between two planes is the shortest distance between them, so the planes that are not parallel, cross each other, and hence their distance is zero. To determine the distance between two parallel planes, we can calculate the length of the perpendicular vector between the surfaces of the two planes. Let us now see the formula to determine the distance between two planes.

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, let us go through the formulas to determine 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 to determine the distance between two parallel lines. As we know, the coordinates of the normal vectors of the two parallel planes are either proportional or equal. So, consider equations of two parallel planes as P₁: ax + by + cz + d₁ = 0 and P₂: ax + by + cz + d₂ = 0. Then, the formula for the distance between two planes that are parallel is given by: |d₂ - d₁|/√(a² + b² + c²). Please note that if the coefficients a, b, c are not equal, then we make them equal using the common ratio a₁/a₂ = b₁/b₂ = c₁/c₂ to get the equivalent equation of the plane.

Distance Between Two Non-Parallel Planes

As we know, the distance between two planes is nothing but the shortest distance between them. So if two planes are not parallel, then they must intersect at some straight line in a three-dimensional space. Hence, the shortest distance between two non-parallel planes is equal to zero. Therefore, the distance between two planes that are not parallel is always zero.

Next, we will explore the method to determine the distance between the two planes using the point-plane distance formula. The formula to calculate the distance between a point (x₁, y₁, z₁) to a plane ax + by + cz + d = 0 is given by D = |ax₁+by₁+cz₁+d |√(a²+b²+c²). To find the distance between two planes using the point-plane distance formula, we can follow the steps given below:

• Step 1: Convert the equations of the two planes into the standard format, i.e., ax + by + cz + d = 0
• Step 2: Check if the planes are parallel. [Two planes P₁: a₁x + b₁y + c₁z + d₁ = 0 and P₂: a₂x + b₂y + c₂z + d₂ = 0 are parallel if a₁/a₂ = b₁/b₂ = c₁/c₂]
• Step 3: Consider the coefficients a, b, c, d from the equation of one of the planes.
• Step 4: Consider a point P(x₁, y₁, z₁) on the other plane. [An easy way to find the point is by taking x = y = 0, and finding the value of z from the equation of the other plane]
• Step 5: Substitute the values of a, b, c, d, x₁, y₁, z₁ into the formula for the distance between a point and plane: |ax₁+by₁+cz₁+d |√(a²+b²+c²)

Following the above steps, we can find the distance between two planes using the formula for distance between point and plane.

Now that we know the two methods to find the distance between two planes, let us solve a few examples based on these methods to understand their application.

Example 1: Find the distance between two planes P₁: 2x + 4y + z + 7 = 0 and P₂: 4x + 8y + 2z - 14 = 0.

Solution: First, we will check if the planes are parallel. Take the ratio of the coefficients in the equations of the two planes. Here, we have a₁ = 2, b₁ = 4, c₁ = 1 and a₂ = 4, b₂ = 8, c₂ = 2. Therefore, we have, a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2 ⇒ The two planes are parallel. Now, to get equal coefficients in the equations of the two planes, divide the equation of P₂ by 2. Then, we have P₂: (1/2)(4x + 8y + 2z - 14) = (1/2) (0) ⇒ P₂: 2x + 4y + z - 7 = 0. Now, we have a = 2, b = 4, c = 1, d₁ = 7, d₂ = -7. Now the formula to find the distance between two planes P₁ and P₂ is: |d₂ - d₁|/√(a² + b² + c²). Therefore, the required distance is,

d = |-7 - 7|/√(2² + 4² + 1²)

= |-14|/√(4 + 16 + 1)

= 14/√(21)

= (2/3)√21 units

Next, let us find the distance between two planes using the point-plane distance formula.

Example 2: Calculate the distance between two planes P₁: 2x + 4y + z + 7 = 0 and P₂: 4x + 8y + 2z - 14 = 0.

Solution: As we checked in example 1, the two planes P₁ and P₂ are parallel. Consider a = 2, b = 4, c = 1, d = 7 from the first plane. Next, we will find a point (x₁, y₁, z₁) on the other plane. For this, assume x₁ = y₁ = 0, and substitute these values in the equation of P₂, we have,

4(0) + 8(0) + 2z₁ - 14 = 0

⇒ 2z₁ - 14 = 0

⇒ z₁ = 7

Therefore, we have a point (0, 0, 7) on plane P₂ and the equation of the first plane P₁: 2x + 4y + z + 7 = 0. Now, we will find the distance between the point (0, 0, 7) and plane P₁: 2x + 4y + z + 7 = 0 using the formula |ax₁+by₁+cz₁+d |√(a²+b²+c²).

d = |ax₁+by₁+cz₁+d |√(a²+b²+c²)

= |2 × 0 + 4 × 0 + 1 × 7 + 7|/√(2² + 4² + 1²)

= 14/√21

= (2/3)√21 units

Hence, we have the same distance between the two planes P₁ and P₂ using the two methods, i.e., (2/3)√21 units

Important Notes Distance Between Two Planes

• The distance between two planes is zero if they are intersecting.
• The distance between two planes P₁: ax + by + cz + d₁ = 0 and P₂: ax + by + cz + d₂ = 0 that are parallel is given by: |d₂ - d₁|/√(a² + b² + c²)
• The distance between two parallel planes can also be calculated using the point-plane distance formula.

Related Topics on Distance Between Two Planes

• Distance Between Two Points
• Euclidean Distance Formula
• Geometry

What is Distance Between Two Planes in Geometry?

The distance between two planes is given by the length of the normal vector that drops from one plane onto the other plane and it can be determined by the shortest distance between the surfaces of the two planes.

How to Find the Distance Between Two Planes?

The distance between two planes can be determined using two methods. We can use the formula |d₂ - d₁|/√(a² + b² + c²) or using the point-plane distance formula.

What is the Formula for the Distance Between Two Planes?

Consider equations of two parallel planes as P₁: ax + by + cz + d₁ = 0 and P₂: ax + by + cz + d₂ = 0. Then, the formula for the distance between two planes that are parallel is given by d = |d₂ - d₁|/√(a² + b² + c²).

What is the Distance Between Two Parallel Planes?

The distance between two parallel planes P₁: ax + by + cz + d₁ = 0 and P₂: ax + by + cz + d₂ = 0 can be determined using the formula d = |d₂ - d₁|/√(a² + b² + c²)

How Do You Find Distance Between Two Non-Parallel Planes?

The distance between two planes is nothing but the shortest distance between them. So if two planes are not parallel, then they must cross each other. Therefore, the distance between two planes that are not parallel is always zero.