Complex Vectors vs Normal Vectors

[jaydnul]

The way I understand it, they both have rectangular forms which are easy for addition/subtraction. Now I realize that the polar form of a complex vector can be simplified into an exponential, which is ideal for multiplication/division.

But this is what confuses me; vectors don't multiply/divide like scalars. So why don't complex vectors have dot products and cross products like normal vectors do?

Example:
(1+2j)*(2+1j)=(\sqrt{5}e^{j63})*(\sqrt{5}e^{j27})=5e^{j90}=(0+5j)

 

[Simon Bridge]

You can:
http://www.solitaryroad.com/c606.html

 

[jaydnul]

So we can treat complex vectors as just normal vectors in the iy by x plane.

But now I'm confused about just the straight multiplication. Are they not being treated like vectors when you just multiply them like complex numbers instead of vectors? That doesn't seem right because you end up with vector in the same plane (which doesn't happen with dot or cross product).

 

[nasu]

You can think about complex numbers as "vectors" in 2D, in the more general meaning of this concept. (see vector spaces in algebra).
However the complex numbers and the 3D vectors we use in physics are not examples of the same object. They have different properties even if you may call them both by the same generic name and represent them by arrows.
The usual cross product is not even defined in 2D.

 

[FactChecker]

But now I'm confused about just the straight multiplication. Are they not being treated like vectors when you just multiply them like complex numbers instead of vectors? That doesn't seem right because you end up with vector in the same plane (which doesn't happen with dot or cross product).

Multiplication of complex numbers gives a basic operation that represents rotations in the XY plane. That leads to the amazing properties of the complex number system and analytic functions. Those properties are very important.

 

[Simon Bridge]

You can represent many things as vectors, spatial coordinates for example. But any ordered list of numbers may be treated as a vector. Just list the numbers, separated by commas, inside parentheses, and it's a vector.

Spacial vectors have a dot product, a cross product, and a normal algebraic product when you write them out as so much i plus so much j but you have to decide how to handle the product of the unit vectors - that part is a bit ambiguous.

Anyway: these different processes tell us things about spatial relationships.

Complex numbers are an intreguing example - if you work out the dot and cross products you'll see they have a relationship with the regular algebraic product ;)
The different processes tell us things about the relationships in the complex plane - they can also be used to tell us about rotations in space if we use them the right way.
The standard algebraic product works for complex numbers because there is no ambiguity about what to do with the product of i with itself (for example) that there is with a unit vector.

Vectors need not have any spatial relationships at all ... i.e. the set of polynomials form a "vector space" and you can represent any polynomial as an ordered list of it's coefficients. So y = ax^2+bx+c = (c,b,a,0,0,0,...). Just like any other vector, polynomial vectors have scalar and cross products etc. It may challenge your intuition to figure out what the different products would mean for this example.

It is very common, in math modelling, to represent a data set by an ordered list of numbers: treated as a vector.
i.e. I could sample y=f(x) for discrete points $\vec x=(x1,x2,x3,\cdots)$ to get discrete results: $\vec y = (y1, y2, y3,\cdots)$ and the effect of the function becomes that of a matrix A that turns the vector x into the vector y: $\vec x = A\vec y$. It can make sense to do this because we don't usually measure things continuously.

How to manipulate "vector spaces" (groups of vectors which have special group properties) is the subject of algebra courses, which occupy years at college level - therefore, a bit big for this post to go into in one go. There are many algebra courses online. ;)

The bottom line, per your question, is that vectors are a mathematical concept which can also be useful in physics. What the vectors mean, what you can do with a specific vector, how you do it, and what that means, all depends on what sort of vectors you are using.
As usual: context is everything.

 

[anorlunda]

May I recommend the following book. I'm sure it will entertain you as it educates you about complex numbers.

https://www.amazon.com/dp/0691146004/?tag=pfamazon01-20

 

[kith]

But now I'm confused about just the straight multiplication. Are they not being treated like vectors when you just multiply them like complex numbers instead of vectors? That doesn't seem right because you end up with vector in the same plane (which doesn't happen with dot or cross product).

In vector spaces, you have a "nice" vector addition in the sense that the sum of two vectors is a vector again and that the operation is associative and commutative. In general, you can't define an equally nice multiplication. You may have a variety of similar operations (like the cross product, the dot product and the scalar multiplication) but all of them lack certain properties.

It is a very peculiar property of the \mathbb{R}^2 that you can define such a nice multiplication. This makes it also a field, the complex numbers. (Note that the property is only peculiar for vector spaces with dimension greater than one. All fields are also one-dimensional vector spaces.)

So your intuition is basically right. The \mathbb{R}^2 is just an exception where we can define a very nice additional structure.