What is cross product of two complex numbers?

[yungman]

Let U=a+jb, V=c+jd.
What is U X V? Where "X" is the cross product. Can you explain?

 

[yungman]

Let U=a+jb, V=c+jd.
What is U X V? Where "X" is the cross product. Can you explain?

 

[rock.freak667]

UxV would give the area of the parallelogram formed by those two complex numbers.

 

[yungman]

UxV would give the area of the parallelogram formed by those two complex numbers.

Can you please write out the steps to get the answer. I am very despirate!

Thanks a million!

 

[rock.freak667]

It basically works out as being

|a b|
|c d|

 

[flatmaster]

I believe you treat the real axis and imaginary axis as two orthogonal directions. Imagine the real component has an implied unit vector sitting out front. Think of the imiganary number j as a unit vector pointing in the imaginary direction

 

[yungman]

It basically works out as being

|a b|
|c d|

You mean U X V = ad-bcj so Re[U X V]=ad, Im[U X V]=bc ?

 

[yungman]

I believe you treat the real axis and imaginary axis as two orthogonal directions. Imagine the real component has an implied unit vector sitting out front. Think of the imiganary number j as a unit vector pointing in the imaginary direction

You mean U X V = ad-bcj so Re[U X V]=ad, Im[U X V]=bc ?

 

[flatmaster]

I believe that's correct. Immagine if you tried to cross two real numbers. You would expect to get exactly 0 which you do.

 

[flatmaster]

What is stil ambiguious to me is that the result of a cross product should still be a vector that is perpendicular to both U and V. What way does this vector point?

 

[flatmaster]

I found another definition. I think it may break down to what I said before

z×w = (z'w - zw')/2.

Where I believe prime indicates a somplez conjagate.

http://www.cut-the-knot.org/arithmetic/algebra/RealComplexProducts.shtml

 

[Defennder]

You mean the cross product of two vectors instead, I believe. On the Argand plane, you can treat the complex numbers as vectors. But then the area of parallelogram formed would be a real number (remember to take the absolute value as well), so you shouldn't have any imaginary part.

 

[yungman]

I found another definition. I think it may break down to what I said before

z×w = (z'w - zw')/2.

Where I believe prime indicates a somplez conjagate.

http://www.cut-the-knot.org/arithmetic/algebra/RealComplexProducts.shtml

Many thanks, I actually paste the whole thing, all the topics of the complex numbers.

 

[yungman]

Thanks everybody.

 

[flatmaster]

There's still something wierd, the source I cyted ended up giving the same intuative result I had as thinking of the real axes and imaginary axes as orthogonal basis, but the real component was still imiginary. IE

z = a+bi
w = c+di

z X w = yadda yadda yadda = adi - bci

 

[flatmaster]

There are still some nice things that are comming out though. For instance, if two vectors point in the same direction in the real - imiginary plane, their cross is zero

z = a+bi
w = c+di


c = 2 a d = 2 b

z X w = adi - bci = a(2b)i - b(2a)i = 0

 

[yungman]

There's still something wierd, the source I cyted ended up giving the same intuative result I had as thinking of the real axes and imaginary axes as orthogonal basis, but the real component was still imiginary. IE

z = a+bi
w = c+di

z X w = yadda yadda yadda = adi - bci

I have been studying up the articles you gave me. The way I understand it is ZXW is pure imaginary.

(Z'W-ZW')/2=i(ad-bc) = i[Im(Z'W)] = i[Im(ZW')]

That is the reason it is called complex product. There is no real part from the calculation.



The real product is Z.W=(Z'W+ZW')/2=ac+bd with no imaginary part.


Thanks for all your help, the link you gave was a good one. I already print out all the chapters and put it in my notes.

Good night

 

[yungman]

Cross product of Complex numbers, is the book wrong or me?

This is the exact equations derivation from the “Field and Wave Electromagnetics” by David K. Cheng which have very very few errors. But this don’t make sense. Before I conclude that this is an error, let me run this by you guys/gals.

Re(A)=(A+A*)/2 Re(B)=(B+B*)/2
Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
= (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
= (1/2) Re(A X B* + A X B). (line 4).

I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.

If you look at (line 3) above,
A X B* = (i/2)[(A*B*) – (A*B)]
A* X B = (i/2)[(AB) – (A*B*)]
A X B = (i/2)[(A*B) – (AB*)]
A* X B* = (i/2)[(AB*) – (A*B)]
The sum of the four terms in (line 3) equal ZERO!

As seen on (line 4), (A X B* + A X B) is pure imaginary. So Re(A X B* + A X B) = ZERO!

Am I missing something?
Thanks

 

[yungman]

Cross product of Complex numbers, is the book wrong or me?

This is the exact equations derivation from the “Field and Wave Electromagnetics” by David K. Cheng which have very very few errors. But this don’t make sense. Before I conclude that this is an error, let me run this by you guys/gals.

Re(A)=(A+A*)/2 Re(B)=(B+B*)/2
Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
= (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
= (1/2) Re(A X B* + A X B). (line 4).

I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.

If you look at (line 3) above,
A X B* = (i/2)[(A*B*) – (A*B)]
A* X B = (i/2)[(AB) – (A*B*)]
A X B = (i/2)[(A*B) – (AB*)]
A* X B* = (i/2)[(AB*) – (A*B)]
The sum of the four terms in (line 3) equal ZERO!

As seen on (line 4), (A X B* + A X B) is pure imaginary. So Re(A X B* + A X B) = ZERO!

Am I missing something?
Thanks

 

[D H]

This is the exact equations derivation from the “Field and Wave Electromagnetics” by David K. Cheng which have very very few errors. But this don’t make sense. Before I conclude that this is an error, let me run this by you guys/gals.

Re(A)=(A+A*)/2 Re(B)=(B+B*)/2
Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
= (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
= (1/2) Re(A X B* + A X B). (line 4).

Kind of roundabout way to get to that result, but this is indeed a tautology for any two complex numbers A and B.

I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.

A*B-AB* is pure real because AB* = (A*B)*. However, your equality is not valid. Show your derivation.

 

[buffordboy23]

There's a lot going on there in your post, so I didn't look at it much. Here's a link that shows various forms for the cross product of two complex numbers--see bottom portion of page 6.

http://books.google.com/books?id=9zKl4lXEXlsC&pg=PA6

You can compare it with your book.

 

[D H]

That stuff in Schaum's Outline is non-standard nomenclature. AxB typically denotes the complex product, not the "cross product". I don't have Field and Wave Electromagnetics, but I would suggest you first check that they aren't discussing the complex product there.

BTW, even if they are using the cross product, the result is still valid because the cross product of two pure real numbers (or two pure imaginary numbers for that matter) is identically zero.

 

[yungman]

Kind of roundabout way to get to that result, but this is indeed a tautology for any two complex numbers A and B.


A*B-AB* is pure real because AB* = (A*B)*. However, your equality is not valid. Show your derivation.

I don't think so, this is the equation:

Let A=a+bi, B=c+di

A X B= (a-bi)(c+di)-(a+bi)(c-di)=[(ac+bd)+i(ad-bc)] - [(ac+bd)+i(bc-ad)]
=i[2(ad-bc)] this is pure imaginary

 

[yungman]

There's a lot going on there in your post, so I didn't look at it much. Here's a link that shows various forms for the cross product of two complex numbers--see bottom portion of page 6.

http://books.google.com/books?id=9zKl4lXEXlsC&pg=PA6

You can compare it with your book.

I have the book, I verify all my formulas already. In fact I got this link from a person here which is much better than Scharms:

http://www.cut-the-knot.org/arithmetic/algebra/RealComplexProducts.shtml

THanks

 

[D H]

That was a stupid typo, sorry. A*B-AB* is obviously pure imaginary. Sorry about that.

Have you verified what Cheng means by the symbol 'x'? I've look through a couple of my complex analysis texts and a couple of my EM texts. None of them use the concept of a complex cross product.

Regardless, the relationship is true for both the complex product and the cross product.

Edit
I think you should double-check the nomenclature in Cheng. Why in the world would you take the real part of an expression that is by definition pure imaginary (the cross product)?

 

[yungman]

That was a stupid typo, sorry. A*B-AB* is obviously pure imaginary. Sorry about that.

Have you verified what Cheng means by the symbol 'x'? I've look through a couple of my complex analysis texts and a couple of my EM texts. None of them use the concept of a complex cross product.

Regardless, the relationship is true for both the complex product and the cross product.

Thanks
The section in question pertain to Poynting vector. The expression is used for P= E X H

I think you got a point while I was typing this, the Poynting vector equation is a true vector equation and is the cross product of E and H. The trick is both E and H are complex also, I got mix up! I have to think about this a little more.

The real equation is:
Re(E)=(E+E*)/2 Re(H)=(H+H*)/2
P = Re(E) X Re(H) = [(E+E*)/2] X [(H+H*)/2]
= (1/4)[(E X H* + E* X H) + (E X H + E* X H*)] (line 3).
= (1/2) Re(E X H* + E X H). (line 4).

 

[Fredrik]

Those first 4 lines are correct. Your confusion is based on the result that AB*+AB is imaginary, right? AB*+AB=A(B*+B)=2A Re(B) is only imaginary if A is.

By the way, this has nothing to do with cross products. This is just multiplication of complex numbers.

 

[yungman]

Those first 4 lines are correct. Your confusion is based on the result that AB*+AB is imaginary, right? AB*+AB=A(B*+B)=2A Re(B) is only imaginary if A is.

By the way, this has nothing to do with cross products. This is just multiplication of complex numbers.

I see what you are talking about AB*+AB=A(B*+B)=2A Re(B) is only imaginary if A is. I was too quick to draw conclusion.

But what I said up to (line 3) is still true. How do you get from (line 3) to (line 4) ? Sum of (line 3) still equal to ZERO regardless of (line 4).



The section in question pertain to Poynting vector. The expression is used for P= E X H

The real equation is:
Re(E)=(E+E*)/2 Re(H)=(H+H*)/2
P = Re(E) X Re(H) = [(E+E*)/2] X [(H+H*)/2]
= (1/4)[(E X H* + E* X H) + (E X H + E* X H*)] (line 3).
= (1/2) Re(E X H* + E X H). (line 4).

 

[Defennder]

This is the 3rd thread you have started on this:

https://www.physicsforums.com/showthread.php?t=276352
https://www.physicsforums.com/showthread.php?t=276350

Kindly stick to only one thread for the same question.

And Fredrik is right. This isn't cross product of a complex number; there is no such thing. Cross product is defined for vectors. And in that book the vectors E,H are actually phasors which means that their x,y,z components may be complex instead of simply being real. So this means you are dealing with cross product of vectors and not complex numbers. The asterisk on the superscript of a bolded letter means that we want the complex conjugate of that vector; change every complex directional component of that vector into into its complex conjugate.

 

[yungman]

This is the 3rd thread you have started on this:

https://www.physicsforums.com/showthread.php?t=276352
https://www.physicsforums.com/showthread.php?t=276350

Kindly stick to only one thread for the same question.

And Fredrik is right. This isn't cross product of a complex number; there is no such thing. Cross product is defined for vectors. And in that book the vectors E,H are actually phasors which means that their x,y,z components may be complex instead of simply being real. So this means you are dealing with cross product of vectors and not complex numbers. The asterisk on the superscript of a bolded letter means that we want the complex conjugate of that vector; change every complex directional component of that vector into into its complex conjugate.

Me bad, I was despirated.

You have the book and know where it is? I am starting to question that also. I have to go back and think about this a little.

So in conclusion, there is NO cross product in complex numbers, X only mean "complex product" which A X B = A*B - AB*. In the book, E and H are vectors in xyz coordinate and with complex value.

Can you show me how to go from (line 3) to (line 4)?

Thanks

 

[D H]

This isn't cross product of a complex number; there is no such thing.

Apparently Schaum's Outlines and others have something they call the cross product for complex numbers. It is highly non-standard, and obviously leads to confusion.

Can you show me how to go from (line 3) to (line 4)?

In the twice-replicated OP, you wrote
Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
= (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
= (1/2) Re(A X B* + A X B). (line 4).

Rearrange line 3:

$$\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 4 \Bigl((A\times B^* + A\times B) + (A^*\times B + A^*\times B^*)\Bigr)$$

The pair of expressions in the inner parentheses on the right hand side are complex conjugates of one another:

$$(A^*\times B + A^*\times B^*) = (A\times B^* + A\times B)^*$$

For any complex number c, c+c^*=2 Re(c). Thus,

$$\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 2 \mathrm{Re}(A\times B^* + A\times B)$$

 

[yungman]

Apparently Schaum's Outlines and others have something they call the cross product for complex numbers. It is highly non-standard, and obviously leads to confusion.



In the twice-replicated OP, you wrote
Re(A) X Re(B) = [(A+A*)/2] X [(B+B*)/2]
= (1/4)[(A X B* + A* X B) + (A X B + A* X B*)] (line 3).
= (1/2) Re(A X B* + A X B). (line 4).

Rearrange line 3:

$$\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 4 \Bigl((A\times B^* + A\times B) + (A^*\times B + A^*\times B^*)\Bigr)$$

The pair of expressions in the inner parentheses on the right hand side are complex conjugates of one another:

$$(A^*\times B + A^*\times B^*) = (A\times B^* + A\times B)^*$$

For any complex number c, c+c^*=2 Re(c). Thus,

$$\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 2 \mathrm{Re}(A\times B^* + A\times B)$$

Thanks a million. I work out the equation and I understand now. I got confused by the books relating cross product with complex product.

Have a nice evening.