3rd type of vector multiplication beside dot/cross product?

I was reading up on how to find the square root of i , and I learned that multiplication of complex numbers could be viewed geometrically by viewing the complex numbers as coordinates on the complex number plane $a_1+b_1i = (a_1,b_1)$ and $a_2+b_2i = (a_2,b_2)$. One can take the polar coordinates of the complex numbers to give $(a_1,b_1) \Rightarrow r_1$, angle = $w$ and $(a_2,b_2) \Rightarrow r_2$, angle = $k$ . And finally the multiplication of the two numbers can be viewed as multiplying $r_1$ and $r_2$, while adding the angles $w$ and $k$, to give the product $r_3= (r_1)(r_2)$, angle = $w+k$ . This can be used to intuitively find the square root of $0 + 1i$.

However in my math textbooks I have not seen any type of vector multiplication similar to this in regards to the real numbers, only dot products and cross products. Does this type of multiplication serve some purpose in regards to real number vectors, does it describe something interesting? Or is it only useful when it comes to multiplying vectors in the complex number plane?

Edit: $r$ is describing the overall length, or magnitude, of the vector. The angle represents the direction the vector is pointing in in regards to the plane it's on.

When doing vector geometry in the plane (pure vector geometry with no coordinate system), all directions “look equal”.

Introducing complex multiplication of such vectors requires a breaking of this symmetry: you must single out a direction to play the role of the positive real axis, so that you can define the polar angle. And then vectors in different directions will behave differently with respect to multiplication. Think of squaring, for example: $1$ maps to itself ($1^2=1$), while $i$ doesn't ($i^2=-1 \neq i$).

The dot and cross products are more geometrical, in the sense that they don't depend on making certain directions special. (Well, actually the cross product is rather weird too, and it's better to learn about the exterior and Clifford products, as recommended in the answer by Bye_World.)

The analogy you’re looking for isn’t any type of vector multiplication per se. It’s a matrix multiplication. Identify the complex number $z=a+bi=re^{i\theta}$ with the matrix $$Z=\pmatrix{a&-b\\b&a}=r\pmatrix{\cos\theta&-\sin\theta\\\sin\theta&\cos\theta},$$ a scaled rotation. Then the product $z_1z_2$ directly maps to the matrix product $Z_1Z_2$. This identification corresponds to the geometric interpretation of multiplication by a complex number as a combination rotation and dilation.

Let $I=e_1\wedge e_2$ be the positively-oriented unit pseudoscalar of the plane. Then $a+bI$ where $a,b\in\Bbb R$ is a spinor$^\dagger$ -- i.e. an object that we use to rotate (and scale) vectors and their higher dimensional analogs. Then the composition of spinors corresponds to multiplication -- where multiplication works exactly as it does with the complex numbers: $$R_1 = a+bI \\ R_2 = c+dI \\ R_2\circ R_1 = (c+dI)(a+bI) = (ca-db)+(cb+da)I$$

Note that we can also represent spinors in polar form: $$R_1 = a+bI = |r_1|e^{\theta_1I} \\ R_2 = c+dI = |r_2|e^{\theta_2I} \\ R_2\circ R_1 = |r_1||r_2|e^{(\theta_1+\theta_2)I}$$

So algebraically, the set of spinors in the plane behave exactly like complex numbers. And, in fact, moving up a dimension, the set of spinors in $3$-space behave exactly like quaternions.

References

For detailed information on the mathematics behind spinors/ rotors, the Wikipedia page on Geometric Algebra is fairly thorough. However more comprehensive references include

• Linear and Geometric Algebra by Alan Macdonald
• Clifford Algebra to Geometric Calculus by David Hestenes & Garret Sobczyk
• Geometric Algebra for Physicists by Chris Doran & Anthony Lasenby

^{$\dagger:$ If $a^2+b^2 = 1$ -- that is, if the spinor is normalized -- then the action of it on a vector is simply to rotate rather than rotate and scale and in this special case they are called rotors.}