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 z1,z2,z3 and θ

.Consider the vector z3−z2.  Let its argument be θ1. Similarly, let the argument of the vector z1−z2  be θ2. Now, a little thought will show you that θ is simply θ1−θ2

.

Now we write z3−zand z1−zin Euler’s form

z3−z2=|z3−z2|e1...(1)

z1−z2=|z1−z2|e2...(2)

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

z3−z2 /z1−z2=|z3−z2|/|z1−z2|e