Angle Between Two Planes

In geometry, the angle between two planes is the dihedral angle, which equals the angle between their normals. You can visualize it by drawing perpendicular lines from the line of intersection in each plane; the angle between those lines is the angle between the two planes. Alternatively, working directly with the normal vectors n₁ and n₂, the acute angle θ between the planes satisfies;
\(\cos \theta = \frac{n_1}{n_2} \)
 

The angle between the two planes is the angle between their normal vectors.
Below are the equations representing the two planes;

Plane 1: \(a_1x + b_1y + c_1z + d_1 = 0 \)
Plane 2: \(a_2x + b_2y + c_2z + d_2 = 0 \)

Now, we extract the normal vector from both planes.

Normal to plane 1: \(\mathbf{n_1} = (a_1,\, b_1,\, c_1) \)
Normal to plane 2:\(\mathbf{n_2} = (a_2,\, b_2,\, c_2) \)

The angle between the planes is the angle between these two vectors.

Use the formula \(\cos \theta = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{|\mathbf{n_1}|\,|\mathbf{n_2}|} \)
Dot product: \(\mathbf{n_1} \cdot \mathbf{n_2} = a_1a_2 + b_1b_2 + c_1c_2 \)

Magnitude:
\(|\mathbf{n_1}| = \sqrt{a_1^2 + b_1^2 + c_1^2} \)
\(|\mathbf{n_2}| = \sqrt{a_2^2 + b_2^2 + c_2^2} \)
 

To calculate the angle between two planes, we use the formula

\(\cos \theta = \frac{n_1 n_2}{|n_1|\,|n_2|} \).

As we have already discovered in the previous section, the dot product is

 \(n_1 n_2 = a_1a_2 + b_1b_2 + c_1c_2 \) and the magnitude is,

\(n_1 = \sqrt{a_1^2 + b_1^2 + c_1^2}, \quad n_2 = \sqrt{a_2^2 + b_2^2 + c_2^2} \).

Substituting these in the formula, \(\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \, \sqrt{a_2^2 + b_2^2 + c_2^2}} \)
 

This is the calculation required to find the angle between two planes in the Cartesian plane.

The angle between two planes is the dot product of the normal vectors of those planes, and can be found using the formula

\(\cos \theta = \frac{n_1 n_2}{|n_1|\,|n_2|} \)

Where,
\(n_1 = (a_1,\, b_1,\, c_1) \)
\(n_2 = (a_2,\, b_2,\, c_2) \)

Angle Between Two Planes in Vector Form
When two planes are written using vector equations, their general form is \(r_n = d \)
Where,

r is the position vector
n is the normal vector of the plane, and
d is a constant.
If,

Plane 1: \(r_{n1} = d_1 \)
Plane 2:\(r_{n2} = d_2 \)

Then the angle between two planes () = the angle between n1and n2.

To determine the angle between two planes, follow the given steps:

1. Find the normals of both planes.
    
2. Compute the dot product of the two normals.
    
3. Find the magnitudes (lengths) of the normals.
    
4. Substitute the magnitude values into the cos θ formula.
    
5. Take inverse cosine (cos⁻¹) to find the angle.
    
6. Always take the absolute value to get an acute angle.
    

The angle between two planes helps determine how they intersect in 3D space. By using the dot product of their normal vectors, you can easily calculate this angle.

• Remember the formula: \(\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}} \)
  
   
• Always identify the normal vectors of both planes before starting the calculation.
  
   
• Simplify all coefficients before substituting them into the formula to reduce mistakes.
  
   
• Keep in mind that this formula is derived from the dot product of the normal vectors.
  
   
• If the dot product is zero, planes are perpendicular; if one normal is a multiple of the other, they are parallel.

The geometric concept of angle between two planes helps understand how different objects fit, move, and interact in a three-dimensional space. Given below are some real-world uses of this concept.
 

• Aircraft navigation and flight control: The angle between the wing and the tail of a plane is calculated for optimal aerodynamic performance. This means that the angle influences how a plane remains stable and in control when air flows over the aircraft. If the angles are incorrect, then the aircraft will be unstable.
  
   
• Structural engineering in architecture: The angle between two planes plays an important role when designing slopes like roofs and ramps or intersecting walls. Engineers calculate the angle to determine load distribution, material strength, and support design.
  
   
• Joint articulation in robotics: In robotic arms and legs, the angle between mechanical joints is studied to understand motion control and to avoid collisions. 
  
   
• Geological analysis: Geologists calculate the angle between layers of rock or sediment to understand the shift in Earth’s crust over time. These angles help identify fault lines, tectonic movements, and predict earthquakes. They also ensure safety in mining and construction.
  
   
• Hull design while building a ship: Architects calculate the angle between different planes of a ship’s hull to minimize drag and improve buoyancy. This results in efficient and fast movement of the ship through water.