[Update] "Degenerating" Vector Multiplication to Complex Numbers

[更新]将向量乘法“退化”到复数 苏剑林 · 2011-02-04 · Mathematics vector complex number

Vectors have two kinds of multiplication: the dot product and the cross product, whose results are called the scalar product and the vector product respectively. In many situations, working with these two operations gives us great convenience (indeed, they were originally abstracted from practical problems). However, quite a few two-dimensional problems are described more concisely using complex numbers. So, in order to combine the strengths of both, it's worth "degenerating" the two vector-multiplication operations down into complex numbers (why "degenerate"? Because vectors can be multi-dimensional — 3D, 4D, and so on — while complex-number operations are only two-dimensional; clearly this is a "degeneration" rather than an "extension" ^_^).

Rules of operation:

Dot product:

General rule: $Z_1 \cdot Z_2=|Z_1||Z_2|\cos(arg\frac{Z_2}{Z_1})$

$$\begin{aligned}1\cdot i=0 \\ i\cdot i=1 \\ \exp(i\theta)\cdot \exp(i\varphi)=\cos(\varphi -\theta) \\ iexp(i\theta)\cdot \exp(i\varphi)=-\sin(\theta-\varphi ) \\ Z_1 \cdot Z_2=Z_1 \bar{Z}_2+Z_2 \bar{Z}_1\end{aligned}$$more

Cross product:

Since the cross product of two 2D vectors always points along the third dimension, we can treat the result of the "cross product" of complex numbers as simply a number.

General rule: $Z_1 \times Z_2=|Z_1| |Z_2| sin(arg\frac{Z_2}{Z_1})$

$$\begin{aligned}1\times i=1 \\ i\times i=0 \\ \exp(i\theta) \times \exp(i\varphi)=\sin(\varphi-\theta ) \\ iexp(i\theta) \times \exp(i\varphi)=-\cos(\theta-\varphi ) \\ Z_1 \times Z_2=(Z_1 \bar{Z}_2-Z_2 \bar{Z}_1)i\end{aligned}$$

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:

$$Z\cdot dZ=|Z| d|Z|$$

$Z*(i dZ)=-(iZ)*dZ=+-|Z|\sqrt{dZ*dZ-(d|Z|)^2}$ (sign yet to be determined)

So now complex numbers turn out to have three kinds of multiplication! Each carries a different meaning. The ordinary complex-number multiplication represents a rotation and scaling, while the dot product and cross product correspond respectively to the cosine and sine of the angle between the two complex numbers.

This is an idea I arrived at while using complex numbers to study periodic orbits of the three-body problem — recorded here for the record!