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=n1n2n1 n2
 

The angle between the two planes is the angle between their normal vectors.
Below are the equations representing the two planes;
Plane 1: a1x + b1y + c1z + d1 = 0
Plane 2: a2x + b2y + c2z + d2 = 0
Now, we extract the normal vector from both planes.
Normal to plane 1: n1=a1, b1, c1
Normal to plane 2: n2=a2, b2,c2 
The angle between the planes is the angle between these two vectors.
Use the formula cos=n1n2n1 n2
Dot product: n1n2= a1a2+b1b2+c1c2
Magnitude:
n1=a12+b12+c12
n2=a22+b22+c22
 

To calculate the angle between two planes, we use the formula cos=n1n2n1 n2.
As we have already discovered in the previous section, the dot product is
 n1n2= a1a2+b1b2+c1c2 and the magnitude is, n1=a12+b12+c12, n2=a22+b22+c22.
Substituting these in the formula, cos =   a1a2+ b1b2+ c1c2a1 2+ b12+ c12  a22 + b22 + c22

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=n1n2n1 n2
Where,
n1=a1, b1, c1
 n2=a2, b2, c2

Angle Between Two Planes in Vector Form

When two planes are written using vector equations, their general form is rn=d
Where,
r is the position vector
n is the normal vector of the plane, and
d is a constant.
If,
Plane 1: rn1=d1
Plane 2: rn2=d2
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 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.