Visualization of complex vectors and dot product

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!

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.

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.

Thanks! Good answers!

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.

Mike_bb said:

Thanks! Good answers!

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/

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?

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.

If I understand you correctly we can obtain set of points if we represent complex vector as complex function. Is it true?

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.

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?

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.

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.

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).

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.

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.