[Maths Class Notes] on Dihedral Angle Pdf for Exam

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.