Angle Between Two Planes

In geometry, it is important to know about various kinds of surfaces. A line is a one-dimensional surface and a space is a three-dimensional surface. However, a plane is a two dimensional surface with zero thickness. Plane is formed by some stack of lines, which are kept aside from each other. Planes play an important part of 3-D geometry. An infinite number of planes can exist in three-dimensional space and in coordinate geometry. Planes have certain special properties which include:

• Any two distinct planes are either parallel or intersect at a line.

• A line may either lie within the plane or intersect the plane at a single point or parallel to the plane. 

• Two lines are parallel to each other if they are perpendicular to the same plane. 

• Two planes are also parallel to each other if they are perpendicular to the same line.

Angle Between the Two Planes Formula

The angle between two planes is defined as the angle between the normal to the two planes. Two planes are referred to as perpendicular if their standard vectors are rectangular. The angle of separation of two intersecting planes is calculated as the angle of separation of normals to both planes. The angle that exists between two vectors can be determined with the help of vector multiplication. 

Let us consider two planes intersecting at an angle θ as shown in the above figure. Let n₁ and n₂ be the normal vectors drawn to the planes. The equation for both the planes is thus given as

\[\bar{r}\].\[\bar{n_{1}}\] = \[d_{1}\]

\[\bar{r}\].\[\bar{n_{2}}\] = \[d_{2}\]

Cosine of angle between two intersecting planes is given as the cosine of the angle between their normals:

This is the angle between two planes formula when normal vectors are given.

Angle between Two Planes Formula in Cartesian System

Consider the angle between two planes, for example in which two planes intersect at an angle θ. In the cartesian system points are labelled on a plane. A plane is formed with two perpendicular lines called the X-axis and the Y-axis, this is called the cartesian or coordinate plane. 

The equations of these two planes are given in the cartesian coordinate system as A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0. In these two equations, A₁, B₁ and C₁ are the direction ratios of normal to the plane described by the equation A₁x + B₁y + C₁z + D₁ = 0 and A₂, B₂ and C₂ are the direction ratios of normal to the plane defined by the equation A₂x + B₂y + C₂z + D₂ = 0. 

Now consider the angle between the normal to the two planes and (A₁, B₁, C₁) and (A₂, B₂, C₂) are the direction ratios of the normal to both the planes in consideration. The cosine of the angle between the two planes is given as:

Solved Examples of Angle Between Two Planes

1. Find the angle between two planes r.(2i - j + k) = 1 and r.(i + k) = -3

Solution: For finding the angle between the two planes r.(2i - j + k) = 1 and r.(i + k) = -3, Using the formula Cosθ = \[\frac{n_{1}.n_{2}}{|n_{1}||n_{2}|}\].
n₁ = 2i - j + k, n₂ = i + k
\[ |n_{1}| = \sqrt{(2^{2} + (-1)^{2} + 1^{2})} = \sqrt{(4 + 1 + 1)} = \sqrt{6} \]
\[ |n_{2}| = \sqrt{(1^{2} + (0)^{2} + 1^{2})} = \sqrt{(1 + 0 + 1)} = \sqrt{2} \]
Scalar product of the normal vectors is, 

n₁ . n₂ = (2i - j + k) . (i + k) = 2 × 1 + (-1) × (0) + 1 × 1 = 2 + 0 + 1 = 3
Substituting the values into the formula,

\[ cos θ = \frac{|(3)|}{(\sqrt{6} . \sqrt{2}})\]
= \[\frac{3}{\sqrt{12}}\]
= \[\frac{\sqrt{3}}{2}\]
\[ ⇒ θ = cos^{-1}(\frac{\sqrt{3}}{2}) \]
= π/6 radians
Therefore the angle between two planes r.(2i - j + k) = 1 and r.(i + k) = -3 is equal to π/6 radians.

2. Determine the angle between two planes P₁: 3x - 6y + 2z = 7 and P₂: 2x + 2y - 2z = 3

Solution: For determining the angle between two planes in cartesian form using the formula cos θ = \[\frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt{A_{1}^{2}+B_{1}^{2}+C_{1}^{2}}.\sqrt{A_{2}^{2}+B_{2}^{2}+C_{2}^{2}}}\]. 

The equations of the planes are P1: 3x - 6y + 2z = 7 and P2: 2x + 2y - 2z = 3. 

Now, A₁ = 3, B₁ = -6, C₁ = 2, A₂ = 2, B₂ = 2, C₂= -2.

Substituting the values into the formula, 

\[ cos θ = \frac{|(3\times 2 + (-6)\times 2 + 2\times (-2))|}{\sqrt{(32 + (-6)2 + 22)} \sqrt{(22 + 22 + (-2)2)}} \]

= \[ \frac{|(6 + (-12) - 4)|}{\sqrt{(9 + 36 + 4)}\sqrt{(4 + 4 + 4)}} \]

= \[ \frac{|-10|}{(\sqrt{49} \sqrt{12})} \]

= \[ \frac{10}{(7\times 2\sqrt{3})} \]

= \[ \frac{5}{7\sqrt{3}} \]

= \[ \frac{5\sqrt{3}}{21} \]

\[ θ = cos^{-1}(\frac{5\sqrt{3}}{21}) \]

Therefore the angle between the two planes P₁: 3x - 6y + 2z = 7 and P₂: 2x + 2y - 2z = 3 is \[ cos^{-1}(\frac{5\sqrt{3}}{21}) \].

3. How to calculate angle between two planes described by the equations 2x + 4y - 4z - 6 = 0 and 4x + 3y + 9 = 0?

Solution: In these equations of the plane, 

A₁ = 2, B₁ = 4, C₁ = - 4, D₁ = - 6

A₂ = 4, B₂ = 3, C₂ = 0, D₂ = 9

Cosθ = \[\frac{A_{1}A_{2}+B_{1}B_{2}+C_{1}C_{2}}{\sqrt{A_{1}^{2}+B_{1}^{2}+C_{1}^{2}}.\sqrt{A_{2}^{2}+B_{2}^{2}+C_{2}^{2}}}\]

Cosθ = \[\frac{|2 \times 4+4 \times 3+(-4) \times 0|}{\sqrt{2^{2}+4^{2}+(-4)^{2}}.\sqrt{4^{2}+3^{2}+0^{2}}}\]

Cosθ = \[\frac{|8+12+0|}{\sqrt{4+16+16}.\sqrt{16+9+0}}\]

Cosθ = \[\frac{20}{\sqrt{36}.\sqrt{25}}\] = \[\frac{20}{6 \times 5}\] = \[\frac{2}{3}\]

θ = \[Cos^{-1}(\frac{2}{3})\] 

4. How to calculate angle between two planes when the direction vectors of normals of the planes are given as n₁ = 2i + 4j - 2k and n₂ = 6i - 8j - 2k. 

Solution:

The coordinates of normal vector n1 is (2, 4, -2)

The coordinates of normal vector n2 is (6, -8, -2)

Cosθ = \[\frac{n_{1}.n_{2}}{|n_{1}||n-{2}|}\]

Cosθ = \[\frac{(2,4,-2)(6,-8,-2)}{|n1||n2|}\]

\[\frac{2\sqrt{39}}{\sqrt{4+16+4}.\sqrt{36+64+4}}\] = \[\frac{2\sqrt{39}}{39}\] = \[\frac{2}{\sqrt{39}}\]

θ = \[Cos^{-1}\frac{2}{\sqrt{39}}\]

Fun Facts

• Point is a dimensionless geometric shape. 

• A plane is formed by a stack of lines arranged side by side.

Conclusion

In this article, we have explored the concept of the angle between two planes in the cartesian system. We have explained a few examples above based on these formulas for your better understanding of the concept.

The angle between two given planes can be determined by the angle present in  between the normals of the two planes. The angle that lies between the two planes is also known as  the dihedral angle. In other words, the dihedral angle between two planes is the angle between the two intersecting planes.