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 do you mean by Dihedral Angle?
A dihedral angle is an angle that can be defined as the angle that occurred when two different planes intersect each other, either directly or indirectly. If the value of the angle shows that it is acute, then we can obtain its obtuse angle value by reducing it from the sum of supplementary angles, which equals 180o.
2. What is the significance of the Dihedral Angle?
Dihedral Angle plays a predominant role in mathematics to solve complex equations in geometry and to find the relationship between two different planes. It also helps to determine the shape of objects and provides stability too.
3. What are the properties of the Dihedral angle?
The properties of the dihedral angle are as follows:
When two planes intersect, we observe two different angles, which are the acute and obtuse angles. The acute angle is called the dihedral angle, which is present between the two planes. The obtuse angle, on the other hand, is found by subtracting the acute angle by 180 degrees.
Sometimes, the two planes do not intersect and, instead, are parallel to each other. In such cases, the dihedral angle of these planes equals zero.
The dihedral angle is also called the interior angle, which is made by the planes in a 3-D shape, e.g., in a polyhedron.
4. Is there any formula to find the Dihedral Angle?
Yes, there is a simple formula using vectors to find the dihedral angle. This can be taken from the vectors of the planes. Then, we will take a normal value from those vectors and substitute all the values into the formula.
5. Is it possible to prove that the planes are parallel using this Dihedral Angle?
Yes, of course. It is very easy to prove that the planes are parallel to each other. If the planes are parallel, then there is no intersection point between them. Thus, if the dihedral angle is zero, the planes are parallel to each other.
6. Differentiate between Dihedral Angle and Torsion Angle.
Both the angles deal with planes themselves, but the difference occurs in finding perspective. Dihedral Angle finds the value of the intersection point, whereas the torsion angle determines the direction of planes, whether it is positive or negative. But both the angels have their own applications and are reliable too.
7. What are the uses of a dihedral angle?
Dihedral angles and their concepts are used widely in the field of mathematics, which help in solving complex geometric equations. It can also help in determining relationships between the cartesian or coordinate planes and also learning about the structure of different shapes. It helps us determine the stability and shape of different objects.
The uses of dihedral angle are as given below:
It plays a significant role in mathematics and even in chemistry for the calculation of protein analysis. It is used for many experiments too.
It helps to find the interior angles in tetrahedrons and polyhedrons.
This angle can be used to prove that two planes are parallel, i.e., when the planes are parallel, this angle is zero degrees.
The angle may be either acute or obtuse, given the intersection point.
It is also used in the aircraft's wings, where the angle increases the stability of the aircraft by making the lower wing of the aircraft fly higher than the higher wing.