For example, considering two complex numbers (representing vectors)
π§=π₯1+ππ¦1, and π§2=π₯2+ππ¦2.
The dot or scalar product of π§1 and π§2 may be defined as
π§1.π§2=π π[conjugate(π§1)π§2]=π₯1π₯2+π¦1π¦2=|π§1||π§2|cos(πΌ),
where πΌ is the angle between π§1 and π§2, lying between 0 and π.
The cross product of π§1 and π§2 may be defined as the vector
|π§1Γπ§2|=(0,0,π₯1π¦2βπ¦1π₯2).
This vector is perpendicular to the complex plane and has the following magnitude:
|π§1Γπ§2|=πΌπ[conjugate(π§1) π§2]=π₯1π¦2βπ¦1π₯2=|π§1| |π§2| sin(πΌ)