What Is a Dihedral Angle Definition Formula Properties and Solved Examples
Generally, an angle occurs when two lines or line segments intersect each other and this may be either acute or obtuse or right angle. But now, we are discussing the Dihedral Angle, which is also an intersection point between two planes.
Thus, the Dihedral Angle may be defined as an angle that occurs when two planes can intersect each other directly or indirectly. These planes are termed as Cartesian planes or coordinates. In other words, we can define dihedral angle as the interior angle which occurs due to the intersection of two Cartesian planes, which help to determine the shape of objects in two dimensions or three dimensions. For the representation of angles, we can use a combination of line segments or two lines. Here, we will discuss the definition, the formula, and the ways in which we can calculate the problems related to the dihedral angle.
The Formula for Calculating Dihedral Angle
We need to calculate the dihedral angle when two Cartesian coordinates or planes intersect each other. Now, we need to derive a formula from the vectors of given planes. If an equation may represent the vectors of a plane,
Say, ax + by + cz + d = 0,
Then the vector is denoted as n. And,
n = (a,b,c).
In the same way, we will take vectors for both the planes and the notations can be taken as \[n_{1}, n_{2}\].
So, normal vectors can be written as
\[n_{1} = a_{1}, b_{1}, c_{1} \]
\[n_{2} = a_{2}, b_{2}, c_{2} \]
Let us say that \[ \Theta\] will be the dihedral angle. Then the formula can be written as
\[Cos \Theta = \frac{n_{1}}{n_{2}}, i.e.,\]
\[Cos \Theta = \frac{n_{1} \times n_{2}}{\sqrt{n_{1}} \times \sqrt{n_{2}}}\]
\[Cos \Theta = \frac{a_{1} a_{2} + b_{1}b_{2} + c_{1}c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2}+c_{2}^{2}} } \]
This is known as the formula for the dihedral angle.
Procedure to Calculate the Dihedral Angle Using this Formula
We need to calculate the dihedral angle, which is the intersection of two planes in geometry, either in two-dimensional or in three-dimensional. For this, we need to follow some sequential steps as given below:
In the first step, we need to determine the values from the figure and represent them in an equation.
Next, we need to denote normal vectors.
Now, calculate the values of the normal vectors.
Finally, substitute all these values into the Dihedral Angle formula.
Then we get the value of the angle between those intersecting planes.
This is the simple procedure we need to follow to calculate the Dihedral Angle. We can understand more clearly by solving certain examples.
Examples to Find Dihedral Angle
Q. If the planes have equations as 3x+y+4z =0 and x+4y+z =0, find the intersecting angle between the planes.
Sol. Given planes are written as
Plane 1, 3x+y+4z =0.
Plane 2, x+4y+z = 0.
By comparing these equations with standard notation, we can take the values as
\[p_{1} =3, q_{1} =1 , r_{1}= 4 \] and
\[ p_{2}=1, q_{2}=4 , r_{2}= 1\]
Then, we need to substitute these values into the formula
\[ Cos \Theta = \frac{(3 \times 1) + ( 1 \times 4) + (4 \times 1)}{\sqrt{(3 \times 3) + ( 1 \times 1) + (4 \times 4)} \sqrt{(1 \times 1) + (4 \times 4) + (1 \times 1)}}\]
\[ = \frac{(3 + 4 + 4 )}{\sqrt{(9 + 1 + 16)} \sqrt{(1 + 16 + 1)}}\]
\[ = \frac{(11)}{\sqrt{26} \sqrt{18}}\]
\[ = \frac{(11)}{\sqrt{468}}\]
= 0.50
Hence, this is the dihedral angle between the given two planes.
Similarly, we can calculate the values of the dihedral angle between different planes.
Scope of Dihedral Angle
Dihedral angle plays a significant role in mathematics as well as chemistry in calculating the analysis of protein. It is also helpful in various experiments.
The Dihedral angle helps to find the interior angle in polyhedra and tetrahedra.
This angle plays a vital role in proving the planes are moving parallelly.
If the angle is zero, then the planes are parallel to each other.
Dihedral angle is either acute or obtuse, based on the intersection point.
Conclusion
Thus, the dihedral angle can be defined as an angle that lies between the intersection of two Cartesian coordinates. This angle helps to solve sums, especially in geometry, which occur very rarely. The notation, formula, and calculation are simple and easy to understand.
The value of angle also helps in various analyses of chemistry. It has a wide scope with various applications. This is a scoring concept for students and experimental tools for mathematicians and science scholars too. As it is a simple formula to understand and use, everyone can practice it perfectly and achieve their target, which is either score, result, or value.
FAQs on Dihedral Angle in Geometry Explained Clearly
1. What is a dihedral angle in geometry?
A dihedral angle is the angle formed between two intersecting planes in three-dimensional space. It is measured along the line of intersection of the planes. In solid geometry, dihedral angles commonly appear in polyhedra such as cubes, pyramids, and prisms. For example, the angle between two adjacent faces of a cube is a dihedral angle.
2. How do you find the dihedral angle between two planes?
The dihedral angle between two planes is found using the angle between their normal vectors. If the normal vectors are n₁ and n₂, then:
cos θ = (n₁ · n₂) / (|n₁||n₂|)
Steps:
- Find the normal vector of each plane.
- Compute their dot product.
- Divide by the product of their magnitudes.
- Take the inverse cosine to get θ.
3. What is the formula for a dihedral angle?
The formula for a dihedral angle between two planes is cos θ = (n₁ · n₂) / (|n₁||n₂|), where n₁ and n₂ are normal vectors. The dot product measures how aligned the planes are. If cos θ = 0, the planes are perpendicular; if cos θ = ±1, the planes are parallel.
4. What is the dihedral angle of a cube?
The dihedral angle of a cube is 90°. Each pair of adjacent faces of a cube meets at a right angle. Since the faces are perpendicular, their normal vectors are also perpendicular, giving cos θ = 0 and θ = 90°.
5. What is the dihedral angle of a regular tetrahedron?
The dihedral angle of a regular tetrahedron is arccos(1/3), which is approximately 70.53°. This angle is formed between any two triangular faces. It is smaller than 90° because the faces tilt inward toward each other.
6. What is the difference between a plane angle and a dihedral angle?
A plane angle is formed by two intersecting lines, while a dihedral angle is formed by two intersecting planes. Key differences:
- Plane angle: 2D concept (measured in a flat surface).
- Dihedral angle: 3D concept (measured between surfaces).
- Plane angles appear in polygons; dihedral angles appear in polyhedra.
7. Can you give an example of calculating a dihedral angle?
Yes, for planes 2x + y − z = 0 and x − y + 2z = 0, the dihedral angle is found using their normal vectors. The normals are n₁ = (2,1,−1) and n₂ = (1,−1,2).
- Dot product: n₁·n₂ = 2(1) + 1(−1) + (−1)(2) = −1
- |n₁| = √6, |n₂| = √6
- cos θ = −1 / 6
8. Why is the dihedral angle important in solid geometry?
The dihedral angle is important because it describes how two faces of a 3D solid are inclined relative to each other. It helps in studying polyhedra, molecular shapes in chemistry, crystallography, and 3D modeling. Understanding dihedral angles allows accurate measurement of shape and structure in space.
9. When is the dihedral angle equal to 90 degrees?
A dihedral angle is 90° when the two planes are perpendicular. This occurs when their normal vectors satisfy n₁ · n₂ = 0. In this case, cos θ = 0 and θ = 90°, meaning the planes intersect at a right angle.
10. How is a dihedral angle measured in practice?
A dihedral angle is measured by taking a cross-section perpendicular to the line of intersection and measuring the angle between the two resulting lines. In coordinate geometry, it is calculated using the dot product formula of normal vectors. In physical models, tools like protractors or digital 3D software are used to measure the angle accurately.
