Visualization of complex vectors and dot product

  • #1
Mike_bb
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3
Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors ##\vec a## and ##\vec b## : ##z=\vec a+ \vec bi##. I know that this complex vector can be visualized in 4D space. But I can't understand how.

2) How is dot product of two complex vectors ##(z_1, z_2) ## possible if ##z_1= \vec a_1 + \vec b_1i## or ##z_2= \vec a_2 + \vec b_2i## consist of two vectors ##\vec a## and ##\vec b##? How does angle between ##z_1## and ##z_2## look like?

Can anyone explain 1) and 2)?
Thanks!
 
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  • #2
Mike_bb said:
Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors ##\vec a## and ##\vec b## : ##z=\vec a+ \vec bi##. I know that this complex vector can be visualized in 4D space. But I can't understand how.
I'm not sure about this. A complex number, ##z##, can be expressed as:
$$z = a + bi$$Where ##a## and ##b## are real numbers.

A complex vector is usually a tuple of complex numbers. For example, a 3D complex vector would be:
$$\vec v = (z_1, z_2, z_3)$$Where ##z_1, z_2, z_3## are all complex numbers.
Mike_bb said:
2) How is dot product of two complex vectors ##(z_1, z_2) ## possible if ##z_1= \vec a_1 + \vec b_1i## or ##z_2= \vec a_2 + \vec b_2i## consist of two vectors ##\vec a## and ##\vec b##?
The dot product (more generally called an inner product) between two complex vectors can be defined as:
$$\vec v \cdot \vec u = z_1w_1^* + z_2w_2^* + z_3w_3^*$$Where ##\vec v = (z_1, z_2, z_3)## and ##\vec u = (w_1, w_2, w_3)## and ##^*## denotes the complex conjugate.
Mike_bb said:
How does angle between ##z_1## and ##z_2## look like?
The angle between two complex vectors can be defined, but I'm not sure how often it's used. The cosine will be a complex number, so it's not going to have a simple visualisation, as far as I know.
 
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  • #3
Mike_bb said:
Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors ##\vec a## and ##\vec b## : ##z=\vec a+ \vec bi##. I know that this complex vector can be visualized in 4D space. But I can't understand how.

2) How is dot product of two complex vectors ##(z_1, z_2) ## possible if ##z_1= \vec a_1 + \vec b_1i## or ##z_2= \vec a_2 + \vec b_2i## consist of two vectors ##\vec a## and ##\vec b##? How does angle between ##z_1## and ##z_2## look like?

Can anyone explain 1) and 2)?
Thanks!
You have two vectors
##\vec a=(a_1+a_2i, a_3+a_4i)##
##\vec b=(b_1+b_2i, b_3+b_4i)##
Those are 2D vectors over complex numbers with dot (=inner=scalar) product defined as (@PeroK described)
##\vec a \cdot \vec b=(a_1+a_2i)(b_1-b_2i)+(a_3+a_4i)(b_3-b_4i)##

I order to visualise them you can replace the complex field with the real field and get
##\vec a=(a_1,a_2, a_3,a_4)##
##\vec b=(b_1,b_2, b_3,b_4)##

Even they look similar those are not the same vectors.
The vector space is different, dimension and the scalar product are different.
I do not know how the angles over the complex field are defined.
Only the lengths of the corresponding vectors and the distances are the same.

Edit: Still those two vector spaces are isometric. They are geometrically the same.

Those are 4D vectors over real numbers with dot (=inner=scalar) product defined as
##\vec a \vec b=a_1b_1+a_2b_2+a_3b_3+a_4b_4##

The lengths of vectors ##\vec a## and ##\vec b## are
##\left\| \vec a \right\|=\sqrt{\vec a \cdot \vec a}=\sqrt{a_1^2+a_2^2+a_3^2+a_4^2}##
##\left\| \vec b \right\|=\sqrt{\vec b \cdot \vec b}=\sqrt{b_1^2+b_2^2+b_3^2+b_4^2}##

The angle ##\alpha## between those two vectors can be calculated as ##\arccos## of ##\cos \alpha##
$$\cos \alpha=\frac {\vec a \cdot \vec b}{\left\| \vec a \right\| \cdot \left\| \vec b \right\|}$$
All linear combinations ##k \vec a+l \vec b## were ##k, l \in \mathbb{R}## are a 2D vector subspace of the 4D space.
You can visualise ##\vec a## and ##\vec b## in the 2D subspace by drawing angle ##\alpha## and the arrows of lengths ##\left\| \vec a \right\|## and ##\left\| \vec b \right\|## on its legs.
 
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  • #4
Thanks! Good answers! :smile:

I think how to visualise ##\vec a=(a_1+a_2i, a_3+a_4i)## in 4D space without using this form: ##\vec a=(a_1,a_2, a_3,a_4)##

If anyone know how to visualize complex vector ##\vec a## in this way then explain me please.
 
  • #5
Mike_bb said:
Thanks! Good answers! :smile:

I think how to visualise ##\vec a=(a_1+a_2i, a_3+a_4i)## in 4D space without using this form: ##\vec a=(a_1,a_2, a_3,a_4)##

If anyone know how to visualize complex vector ##\vec a## in this way then explain me please.
If you have a complex function of a complex variable, then you can visualise the real and imaginary parts as separate 2D surfaces in 3D space. In your case, you could plot ##a_3## and ##a_4## separately as functions of ##(a_1, a_2)##.

Wolfram Alpha offers some tools, for example:

https://www.wolfram.com/language/12/complex-visualization/
 
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  • #6
PeroK said:
In your case, you could plot a3 and a4 separately as functions of (a1,a2).
Do you propose to represent complex vector as complex function?
 
  • #7
Mike_bb said:
Do you propose to represent complex vector as complex function?
Each particular complex vector would be a point in 3D space representing its real part and a point in another 3D space representing its imaginary part. The question is what do you want to do by visualising a vector? An individual point by itself is pretty boring and hardly needs visualisation. A set of points is different - that could be interesting.
 
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  • #8
If I understand you correctly we can obtain set of points if we represent complex vector as complex function. Is it true?
 
  • #9
Mike_bb said:
If I understand you correctly we can obtain set of points if we represent complex vector as complex function. Is it true?
No. It's just a point in 4D. There's nothing to visualise.
 
  • #10
PeroK said:
No. Why visualise a vector? It's just a point in 4D. There's nothing to see.
In 4D space with coordinate system XYZW (X,Y - real axis, Z, W - imaginary axis) we can obtain point. Is it right?
 
  • #11
PeroK said:
No. It's just a point in 4D. There's nothing to visualise.
I mean that we can obtain coordinate that we need if we use functions. For example, for ##\vec a=(x,2x)## we can use ##y=2x## and we'll obtain set of points.
 
  • #12
Mike_bb said:
I mean that we can obtain coordinate that we need if we use functions. For example, for ##\vec a=(x,2x)## we can use ##y=2x## and we'll obtain set of points.
I don't understand what you're trying to do.
 
  • #13
PeroK said:
I don't understand what you're trying to do.
We can obtain point in coordinate system if we use functions (as you mentioned above) for visualise points (and surface consists of this points).
 
  • #14
Mike_bb said:
We can obtain point in coordinate system if we use functions (as you mentioned above) for visualise points (and surface consists of this points).
That's not what I said. I said a single point is a boring thing to visualise. It's more interesting to think about complex functions of a complex variable.
 
  • #15
PeroK said:
That's not what I said. I said a single point is a boring thing to visualise. It's more interesting to think about complex functions of a complex variable.
You didn't understand me and I didn't understand you)) I mean that we can obtain point on plot if we'll consider vector or function. In our case, we use function for this purpose. For example, ##\vec a(x,2x)## we can represent as function ##y=2x## and we can see on the plot that point (1,2) and (2,4) and so on satisfies this vector.
 

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