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.