PS_1.2.1 Mathematical Expression of Dihedral Angle:

The dihedral angle ($\phi$) is the angle between two planes.

First of all, we should know the General equation of Plane:

$\mathbf{Ax + By + Cz + D = 0}$ .......... (i)

Where $A, B, C, D$ are constants.

And, Equation of plane passing through points $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$ is,

$\begin{vmatrix}
x-x_1 & y-y_1  & z-z_1\\
x_2-x_1 & y_2-y_1 & z_2-z_1\\
x_3-x_1 & y_3-y_1 & z_3-z_1
\end{vmatrix}=0$ .......... (ii)

Or,

$\begin{vmatrix}
y_2-y_1 & z_2-z_1\\
y_3-y_1 & z_3-z_1
\end{vmatrix} (x-x_1) + \begin{vmatrix}
z_2-z_1 & x_2-x_1\\
z_3-z_1 & x_3-x_1
\end{vmatrix} (y-y_1) + \begin{vmatrix}
x_2-x_1 & y_2-y_1\\
x_3-x_1 & y_3-y_1
\end{vmatrix} (z-z_1) = 0$ 

.......... (iii)

After Solving $eq^n (iii)$, we get; 

$\left.\begin{matrix}
\left \{ (y_2-y_1)(z_3-z_1) - (z_2-z_1)(y_3-y_1)\right \}x\;+\\
\left \{ (x_3-x_1)(z_2-z_1) - (x_2-x_1)(z_3-z_1)\right \}y\;+ \\
\left \{ (x_2-x_1)(y_3-y_1) - (x_3-x_1)(y_2-y_1)\right \}z + D = 0 \;\;\;\;\;\;\;\;\;\;\;
\end{matrix}\right\}$ .......... (iv)

Where $D$ is Constant term. This $eq^n (iv)$ gives the equation of plane passing through points.

Now,

From the above references, we have to calculate the Dihedral angle between the two planes:

$a_1x + b_1y + c_1z + d_1 = 0$ ..........  (v)

$a_2x + b_2y + c_2z + d_2 = 0$ ..........   (vi)

Which have normal vectors $n_1 = (a_1, b_1, c_1)$ and $n_2 = (a_2, b_2, c_2)$ is simply given dihedral angle through the dot product of the normals.

i.e. $Cos\phi = \hat{n_1}.\hat{n_2}$   ..........  (vii)

Or, 

$Cos\phi = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$  ..........  (viii)

Note: Unit Vector - If A is a vector with magnitude A $\neq 0$), then $\hat{n} = \frac{\vec{A}}{\left | A \right |}$ is a unit vector having the same direction as $\vec{A}$.

Click here for: Basic concept on Dihederal angle.

PS_1.2 Dihedral Angle:

"Act as if what you do makes a difference. It does."    ~ William James

A dihedral angle:  - angle between two intersecting planes.
In chemistry: - angle between planes through two sets of three atoms, having two atoms in common.
In solid Geometry: - union of a line and two half-planes that have this line as a common edge.

Look around the room we are in one stationary position. And we observe where one of the walls of the room meets the ceiling. There we see, both of these are 2-dimensional flat surfaces which makes them planes.

Notice that where this happens, an angle is formed between two planes. That angle is called a dihedral angle [1].

[Note:The dihedral angle can be defined as the angle through which plane $A$ must be rotated (about their common line of intersection) to align it with plane $B$. For precision, one should specify the angle or its supplement, since both rotations will cause the plane to coincide.]

Picture Source

Given two plane; the measure of an angle formed by intersecting the two planes with another plane orthogonal (the dot product of two vector is Zero i.e. $A.B = 0$) to the line of intersection [2].

Picture Source

The angle measure between the normal direction of the two planes is the same as the measure of dihedral angles.

So, dihedral angle can be measured by taking dot product of the normal directions and using Cosine Theorem for Dot products.

Dihedral angles are used to specify the molecular conformation. 

This is the demonstration of dihedral effect in Aircraft Flight. 

How to I calculate the Dihedral angle?

This is a calculator to calculate the dihedral angle.

Click here for: Mathematical expression of Dihedral angle.