# Rotation theorem

# Rotation theorem in Complex Numbers

Rotational theorem *i.e., *angle between two intersecting lines. This is also known as coni method.

Consider a configuration of complex numbers as shown below:

We know the angle θ

. Our purpose is to write down an expression that relates all the four quantities z_{1},z_{2},z_{3} and θ

.Consider the vector z_{3}−z_{2}. Let its argument be θ1. Similarly, let the argument of the vector z_{1}−z_{2}_{ } be θ2. Now, a little thought will show you that θ is simply θ1−θ2

.

Now we write z_{3}−z_{2 }and z_{1}−z_{2 }in Euler’s form

z_{3}−z_{2}=|z_{3}−z_{2}|e^{iθ}^{1}...(1)

z_{1}−z_{2}=|z1−z2|e^{iθ}^{2}...(2)

Since we know θ1−θ2, we divide (1) by (2) to get

z_{3}−z_{2 }/z_{1}−z_{2}=|z_{3}−z_{2}|/|z_{1}−z_{2}|e^{iθ}