Rotation Of Complex Numbers - Practice Questions & MCQ

Vector Notation

Let us take any complex number $z=x+i y$ so point P(x,y) represents it on the Argand Plane. Then OP can be represented as vector $\overrightarrow{O P}=x \hat{i}+y \hat{j}$, where $\hat{i}$ represents the x-axis while $\hat{j}$ represents the y-axis and O is the origin.

$|\overline{O P}|=\sqrt{x^2+y^2}=|z|$

Therefore complex number z can be represented as  $\overrightarrow{O P}$

Similarly, a vector starting from point A (z₁) and ending at B(z₂) is represented by AB vector which equals (z₂ - z₁)

The length of AB is given by the modulus of this vector $\left|z_2-z_1\right|$

Rotation Theorem (Coni Method)

Let three points A, B and C in the Argand Plane whose affixes are z₁, z₂ and z₃ respectively.

If we rotate AB to AC, then 

$\frac{z_3-z_1}{z_2-z_1}=\frac{\left|z_3-z_1\right|}{\left|z_2-z_1\right|} e^{i \theta}$

Note: The final vector should be in the numerator and the starting vector in the denominator.  is positive if rotation is anti-clockwise and negative if it is clockwise.