Modulus of Sum and Difference

Modulus of Sum and Difference

## Formula

$$|z_1+z_2|^2=|z_1|^2+|z_2|^2+2\operatorname{Re}(z_1\bar z_2)$$

$$|z_1-z_2|^2=|z_1|^2+|z_2|^2-2\operatorname{Re}(z_1\bar z_2)$$

## Special Case

If:

$$\operatorname{Re}(z_1\bar z_2)=0$$

then:

$$|z_1+z_2|^2=|z_1-z_2|^2=|z_1|^2+|z_2|^2$$

## Usage

- Used to simplify modulus expressions and establish geometric relationships.