Introduction
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
1. What is a dihedral angle in three-dimensional geometry?
A dihedral angle is the angle formed by the intersection of two planes in 3D space. Imagine opening a book; the angle between the two pages is a dihedral angle. It is measured by taking a point on the line of intersection and drawing a perpendicular line to it in each plane. The angle between these two perpendicular lines is the dihedral angle.
2. How do you find the dihedral angle using the equations of two planes?
To find the dihedral angle (θ) between two planes, you use the dot product of their normal vectors. If the plane equations are a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0, their normal vectors are n₁ = (a₁, b₁, c₁) and n₂ = (a₂, b₂, c₂). The formula, as per the CBSE 2025-26 syllabus, is:
cos(θ) = |(a₁a₂ + b₁b₂ + c₁c₂)| / (√(a₁² + b₁² + c₁²) * √(a₂² + b₂² + c₂²))
This is equivalent to cos(θ) = |n₁ · n₂| / (||n₁|| ||n₂||).
3. What is the step-by-step method to calculate the dihedral angle from plane equations?
To correctly calculate the dihedral angle between two planes, follow these steps:
- Step 1: Identify the normal vectors, n₁ and n₂, from the coefficients of x, y, and z in the two plane equations.
- Step 2: Calculate the dot product of the two normal vectors: n₁ · n₂.
- Step 3: Calculate the magnitude (or length) of each normal vector: ||n₁|| and ||n₂||.
- Step 4: Substitute these values into the formula cos(θ) = |n₁ · n₂| / (||n₁|| ||n₂||).
- Step 5: Solve for θ by taking the inverse cosine (arccos) of the calculated value.
4. What is the importance of the dihedral angle being 0° or 90°?
The value of a dihedral angle gives important information about the orientation of the planes:
- If the dihedral angle is 0 degrees, it means the normal vectors are parallel, and therefore the two planes are parallel to each other and will never intersect.
- If the dihedral angle is 90 degrees, the normal vectors are perpendicular. This signifies that the two planes are perpendicular or orthogonal to each other.
5. What is the difference between a dihedral angle and the angle between two lines?
The primary difference is the dimension of the objects involved. An angle between two lines measures the rotation between two one-dimensional lines within a plane. A dihedral angle, however, measures the angle between two two-dimensional planes in three-dimensional space. It describes the orientation of surfaces relative to each other, a key concept in 3D geometry.
6. Can you provide a real-world example of a dihedral angle?
A classic real-world application of a dihedral angle is found in aeronautical design. The wings of an airplane are often attached to the main body at a slight upward angle. This angle, formed between the plane of the wing and the horizontal plane, is a dihedral angle. This design feature provides crucial roll stability, helping the aircraft to naturally level itself out after being tilted by wind or turbulence.
7. Is the dihedral angle always considered the acute angle between planes?
Yes, by convention in most mathematical contexts, the dihedral angle refers to the acute angle (the angle ≤ 90°) between two intersecting planes. When two planes intersect, they form both an acute and an obtuse angle that add up to 180°. The standard formula using the absolute value of the dot product, |n₁ · n₂|, is specifically designed to yield this acute angle. If the obtuse angle is required, you can subtract the calculated acute angle from 180°.
8. How does the concept of a dihedral angle apply to polyhedra?
In a polyhedron, such as a cube, pyramid, or prism, the dihedral angle is the internal angle between any two adjacent faces that share a common edge. For instance, in a regular cube, the dihedral angle between any two connected faces is exactly 90°. For a regular tetrahedron, the dihedral angle between any two faces is approximately 70.53°. This angle is crucial for defining the shape and structural properties of these 3D figures.
