What is the Angle Between Two Planes?

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Discover the concept of the Angle between Two planes in three-dimensional space. Explore how to calculate this angle and gain insights into its geometric significance.

What is the Angle Between Two Planes?

The angle between two planes in three-dimensional space is the angle formed by their normal vectors. The normal vector of a plane is a vector that is perpendicular (orthogonal) to the plane’s surface. To find the angle between two planes, you can use the dot product formula and trigonometric functions.

Let’s say you have two planes with normal vectors n₁ and n₂. The angle (θ) between these two planes can be found using the following formula:

• cos(θ) = (n₁ · n₂) / (||n₁|| * ||n₂||)

Where:

n₁ · n₂ is the dot product of the normal vectors.

||n₁|| and ||n₂|| are the magnitudes (lengths) of the normal vectors.

Once you have the cosine of the angle, you can use inverse cosine (arccos) to find the actual angle:

• θ = arccos((n₁ · n₂) / (||n₁|| * ||n₂||))

Keep in mind that the result of arccos will be in radians. If you need the angle in degrees, you can convert it using the fact that π radians = 180 degrees.

It’s important to note that the angle between two planes is always positive and is measured between 0 and 180 degrees (0 and π radians). If the two planes are parallel, the angle between them is 0 degrees (or π radians). If the planes are perpendicular, the angle between them is 90 degrees (or π/2 radians).

Remember that in order to use this formula, you need to know the normal vectors of the two planes. If you only have the equations of the planes, you’ll need to find their normal vectors first.

How to Find the Angle Between Two Planes?

To find the angle between two planes, you can follow these steps:

• Determine the Normal Vectors: The normal vectors of the planes are crucial for calculating the angle between them. You need to find the normal vectors of both planes.
• Calculate the Dot Product: Once you have the normal vectors, calculate the dot product of the two vectors.
• Find the Magnitudes: Calculate the magnitudes (lengths) of the normal vectors.
• Apply the Cosine Formula: Use the dot product and magnitudes to calculate the angle between the planes using the cosine formula.

Here’s a detailed breakdown of each step:

Step 1: Determine the Normal Vectors:

Let’s say you have two plane equations in the form Ax + By + Cz + D = 0. The normal vector of a plane is the vector (A, B, C).

For Plane 1: A1x + B1y + C1z + D1 = 0, the normal vector is (A1, B1, C1).

For Plane 2: A2x + B2y + C2z + D2 = 0, the normal vector is (A2, B2, C2).

Step 2: Calculate the Dot Product:

Calculate the dot product of the two normal vectors:

Dot Product = (A1 * A2) + (B1 * B2) + (C1 * C2).

Step 3: Find the Magnitudes:

Calculate the magnitudes (lengths) of the normal vectors:

Magnitude of Normal Vector 1 = √(A1^2 + B1^2 + C1^2).

Magnitude of Normal Vector 2 = √(A2^2 + B2^2 + C2^2).

Step 4: Apply the Cosine Formula:

Using the dot product and magnitudes, you can calculate the angle (θ) between the planes using the arccosine (inverse cosine) function:

Angle (θ) = arccos((Dot Product) / (Magnitude of Normal Vector 1 * Magnitude of Normal Vector 2)).

Keep in mind that the angle obtained from the above formula will be between 0 and 180 degrees. If you want the acute angle between the planes, you can directly use the calculated angle. If you want the obtuse angle, subtract the calculated angle from 180 degrees.

Remember to work with consistent units and ensure your normal vectors are properly normalised (have a magnitude of 1) if needed.

Note that this method assumes that the planes are not parallel. If the planes are parallel, the angle between them is 0 degrees. If the planes are coincident, the angle is also 0 degrees.

What is the Formula of Angle Between Two Planes?

The formula to calculate the angle between two planes is based on the dot product of their normal vectors. The normal vectors are perpendicular to the planes and help determine their orientations in space. The formula is given by:

• cos(θ) = |n₁·n₂| / (||n₁|| * ||n₂||)

Where:

θ is the angle between the two planes.

n₁ and n₂ are the normal vectors of the two planes.

|n₁·n₂| represents the dot product of the normal vectors.

||n₁|| and ||n₂|| are the magnitudes (lengths) of the normal vectors.

Once you calculate the cosine of the angle (θ), you can find the actual angle by taking the inverse cosine (arccos) of the cosine value.

Keep in mind that angles are usually considered acute (less than 90 degrees) since two planes can never have an angle greater than 90 degrees between them. If you calculate an obtuse angle (greater than 90 degrees), you likely made an error in the calculation or need to use the complementary angle (90 degrees minus the obtained angle) to find the acute angle between the planes.

Angles Between Two Plane in Vector Form

The angle between two planes can be calculated using their normal vectors. If you have the vector representations of the normal vectors of the planes, you can use the dot product formula to find the angle between them.

Let’s say you have two planes given by their normal vectors:

Plane 1: Normal vector A = (a₁, a₂, a₃)

Plane 2: Normal vector B = (b₁, b₂, b₃)

The formula to find the angle θ between these two normal vectors is given by:

cos(θ) = (A · B) / (||A|| * ||B||)

Where:

A · B is the dot product of vectors A and B.

||A|| and ||B|| are the magnitudes (lengths) of vectors A and B, respectively.

Once you have the cosine of the angle, you can find the angle θ using the inverse cosine function (arccos):

θ = arccos((A · B) / (||A|| * ||B||))

Keep in mind that the angle calculated using this formula is the acute angle between the two planes. If you need the obtuse angle, you can subtract the acute angle from 180 degrees.

Remember to ensure that the normal vectors are normalized (have a length of 1) before using the formula to get an accurate angle.

Please note that the above formula assumes that the normal vectors are non-zero and that the planes are not parallel. If the normal vectors are zero or the planes are parallel, the formula may not give a valid result. In such cases, you may need to handle special cases separately.

Also, make sure that your vectors are in a consistent coordinate system when calculating the dot product and magnitudes. If your vectors are not already normalized, you can normalize them by dividing each component by the magnitude of the vector.

Angles Between Two Plane in Cartesian Form

The angle between two planes can be determined using their normal vectors. The normal vector of a plane is a vector that is perpendicular to the plane. Given two planes with normal vectors n1 and n2, the angle θ between them can be calculated using the dot product formula:

cos(θ) = (n1 · n2) / (||n1|| * ||n2||),

where:

n1 · n2 represents the dot product of the normal vectors.

||n1|| and ||n2|| represent the magnitudes (lengths) of the normal vectors.

Once you calculate the cosine of the angle, you can find the angle θ using the inverse cosine function (arccos):

θ = arccos((n1 · n2) / (||n1|| * ||n2||)).

Keep in mind that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them:

n1 · n2 = ||n1|| * ||n2|| * cos(θ).

If you have the Cartesian equations of the planes and need to find their normal vectors, you can convert the equations to their normal form (Ax + By + Cz + D = 0), where (A, B, C) are the coefficients of the normal vector. Then, the normal vector n = (A, B, C).

Once you have the normal vectors of both planes, you can proceed with the calculations outlined above to find the angle between the two planes.

Solved Examples on Angle Between Two Planes

Here we have provide you with some solved examples on finding the angle between two planes.

Example 1:

Find the angle between the planes given by the equations:

Plane A: 2x – 3y + 4z = 5

Plane B: x + 2y – z = 7

Solution:

Step 1: Find the normals of both planes.

The normal vector of Plane A is [2, -3, 4], and the normal vector of Plane B is [1, 2, -1].

Step 2: Use the dot product formula to find the angle θ between the normal vectors.

The dot product of the two normal vectors is:

[2, -3, 4] ⋅ [1, 2, -1] = (2)(1) + (-3)(2) + (4)(-1) = 2 – 6 – 4 = -8

The magnitudes of the normal vectors are:

|n(A)| = √(2^2 + (-3)^2 + 4^2) = √29

|n(B)| = √(1^2 + 2^2 + (-1)^2) = √6

The angle between the planes is given by the formula:

cos(θ) = (n(A) ⋅ n(B)) / (|n(A)| ⋅ |n(B)|)

cos(θ) = (-8) / (√29 ⋅ √6)

θ = cos^(-1)(-8 / (√29 ⋅ √6))

Using a calculator, θ ≈ 95.94 degrees.

Example 2:

Find the angle between the planes defined by the equations:

Plane P: 3x – y + 2z = 4

Plane Q: 2x + 3y – z = 5

Solution:

Step 1: Find the normals of both planes.

The normal vector of Plane P is [3, -1, 2], and the normal vector of Plane Q is [2, 3, -1].

Step 2: Use the dot product formula to find the angle θ between the normal vectors.

The dot product of the two normal vectors is:

[3, -1, 2] ⋅ [2, 3, -1] = (3)(2) + (-1)(3) + (2)(-1) = 6 – 3 – 2 = 1

The magnitudes of the normal vectors are:

|n(P)| = √(3^2 + (-1)^2 + 2^2) = √14

|n(Q)| = √(2^2 + 3^2 + (-1)^2) = √14

The angle between the planes is given by the formula:

cos(θ) = (n(P) ⋅ n(Q)) / (|n(P)| ⋅ |n(Q)|)

cos(θ) = (1) / (√14 ⋅ √14)

θ = cos^(-1)(1 / (√14 ⋅ √14))

Using a calculator, θ ≈ 45 degrees.

These are two examples demonstrating how to find the angle between two planes using their normal vectors and the dot product formula. Remember to use trigonometric functions to find the actual angle from the cosine value.

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