Vectors have two types of multiplication: the dot product and the cross product, whose results are also known as the scalar product and the vector product, respectively. In many cases, using these two defined multiplication operations can provide great convenience (in fact, they are abstracted from practical problems). However, a considerable number of two-dimensional problems are more concisely described using complex numbers. Therefore, to integrate the advantages of both, it is necessary to “degenerate” the two multiplication operations of vectors into complex numbers. (Why use the term “degenerate”? Because vectors are multi-dimensional—they can be 3D, 4D, etc.—while complex number operations are only two-dimensional. Clearly, this is a “degeneration” rather than an “extension” ^_^)
Operational Rules:
Dot Product:
General Rule: Z_1 \cdot Z_2 =
|Z_1||Z_2|\cos\left(\arg\frac{Z_2}{Z_1}\right) \begin{aligned}
1 \cdot i &= 0 \\
i \cdot i &= 1 \\
\exp(i\theta) \cdot \exp(i\varphi) &= \cos(\varphi - \theta) \\
i \exp(i\theta) \cdot \exp(i\varphi) &= -\sin(\theta - \varphi) \\
Z_1 \cdot Z_2 &= Z_1 \bar{Z}_2 + Z_2 \bar{Z}_1
\end{aligned}
Cross Product:
Since the cross product of two-dimensional vectors always points toward
the third dimension, the result of the cross product of complex numbers
can be considered a single scalar value.
General Rule: Z_1 \times Z_2 = |Z_1| |Z_2|
\sin\left(\arg\frac{Z_2}{Z_1}\right) \begin{aligned}
1 \times i &= 1 \\
i \times i &= 0 \\
\exp(i\theta) \times \exp(i\varphi) &= \sin(\varphi - \theta) \\
i \exp(i\theta) \times \exp(i\varphi) &= -\cos(\theta - \varphi) \\
Z_1 \times Z_2 &= (Z_1 \bar{Z}_2 - Z_2 \bar{Z}_1)i
\end{aligned}
Transformation Relations:
\begin{aligned} Z_1 \times Z_2 &= -Z_2 \times Z_1 \\ Z_1 \times (i Z_2) &= Z_1 \cdot Z_2 \\ Z_1 \cdot (i Z_2) &= -Z_1 \times Z_2 \end{aligned}
Differential Identities:
Z \cdot dZ = |Z| d|Z| Z \cdot (i dZ) = -(iZ) \cdot dZ = \pm |Z|\sqrt{dZ \cdot dZ - (d|Z|)^2} \quad \text{(Sign to be determined)}
Consequently, complex numbers now have three types of multiplication. They each represent different meanings! The original multiplication of complex numbers represents a rotation and scaling, while the dot product and cross product are related to the cosine and sine of the angle between two complex numbers, respectively.
This is an insight gained while using complex numbers to study periodic orbits in the three-body problem, recorded here for reference!