How Do We Determine the 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.
FAQs on Angle Between Two Planes
1. What is the concept of angle between two planes?
The angle between two planes is basically equal to the angle between the normal vectors to the two planes. The angle between two planes in vector form and cartesian form is always equal to the angle between the normal to the two planes. The angle formed between two intersecting planes is called a dihedral angle and in other words, the angle between the normal vectors of two planes is called the dihedral angle.
2. How to find angle between two planes?
To find the angle between two planes it is important to determine the angle between the normal vectors to the two planes. By using the following formulas, the angle between two planes can be found. The equations for finding the angle of the two planes in vector form are
\[\bar{r}\].\[\bar{n_{1}}\] = \[d_{1}\] and \[\bar {r}\].\[\bar{n_{2}}\] = \[d_{2}\]
And the equations for finding the angle of the two planes in the cartesian form are
A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0.
The formulas for finding the angle θ between two planes are:
Cosθ = \[\frac{n_{1}.n_{2}}{|n_{1}||n_{2}|}\]
cos θ = \[ \frac{|(A_{1}A_{2} + B_{1}B_{2} + C_{1}C_{2})|}{\sqrt{(A_{1}^{2}+B_{1}^{2}+C_{1}^{2})}\sqrt{(A_{2}^{2}+B_{2}^{2}+C_{2}^{2})}} \]
3. What does the dihedral angle mean?
A dihedral angle is defined as the angle that is between two intersecting planes. In Chemistry, it refers to the angle which is between planes through two sets of three atoms, which have two atoms in common while in geometry, it is defined as the union of a line and two half-planes that are having this line as a common edge. The main functionality of the dihedral effect is to give stability in the roll axis and is a significant factor in the constancy of the spiral model that is occasionally referred to as roll stability.
4. Explain some properties of Planes?
There are various properties of planes like
Any two distinct planes are either considered to be parallel or intersect at a line.
A-line can also lie within the plane or intersect the plane at a single point or parallel to the plane.
Two lines are parallel to each other only when they are perpendicular to the same plane.
Two planes are also parallel to each other only when they are perpendicular to the same line.
5. How are geometric surfaces categorized based on their dimensions?
Geometric surfaces may have one, two or three dimensions.
A point is a dimensionless geometric surface.
A one-dimensional geometric surface is called a line. The line has only one dimension which is called length.
Plane is a two-dimensional geometric surface. The two dimensions of a plane are length and width.
A three-dimensional geometric surface is called space. The three dimensions of measuring a space are length, width, and height.