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yungman
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Let U=a+jb, V=c+jd.
What is U X V? Where "X" is the cross product. Can you explain?
What is U X V? Where "X" is the cross product. Can you explain?
rock.freak667 said:UxV would give the area of the parallelogram formed by those two complex numbers.
rock.freak667 said:It basically works out as being
|a b|
|c d|
flatmaster said: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
flatmaster said: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
flatmaster said: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
Kind of roundabout way to get to that result, but this is indeed a tautology for any two complex numbers A and B.yungman said: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).
A*B-AB* is pure real because AB* = (A*B)*. However, your equality is not valid. Show your derivation.I verified that A X B = (i/2)[(A* B) – (AB*)] which is pure imaginary.
D H said: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.
buffordboy23 said: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 said: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.
Fredrik said: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.
Defennder said: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.
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.Defennder said:This isn't cross product of a complex number; there is no such thing.
yungman said:Can you show me how to go from (line 3) to (line 4)?
D H said: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:
[tex]\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 4 \Bigl((A\times B^* + A\times B) + (A^*\times B + A^*\times B^*)\Bigr)[/tex]
The pair of expressions in the inner parentheses on the right hand side are complex conjugates of one another:
[tex](A^*\times B + A^*\times B^*) = (A\times B^* + A\times B)^*[/tex]
For any complex number c, c+c*=2 Re(c). Thus,
[tex]\mathrm{Re}(A)\times \mathrm{Re}(B) = \frac 1 2 \mathrm{Re}(A\times B^* + A\times B)[/tex]
The cross product of two complex numbers is a mathematical operation that combines two complex numbers to form a new complex number. It is also known as the vector product or outer product.
The cross product of two complex numbers, z1 and z2, is calculated by multiplying the real parts of the two numbers and subtracting the product of their imaginary parts. The result is a new complex number, z1 x z2 = (a1b2 - b1a2) + (a1b2 + a2b1)i, where a and b represent the real and imaginary parts respectively.
The cross product of two complex numbers results in a new complex number, while the dot product of two complex numbers results in a real number. The cross product is also known as an outer product, while the dot product is known as an inner product.
The cross product of complex numbers follows the same properties as vector cross product, such as distributivity, commutativity, and associativity. It also follows the rule of anti-commutativity, which means that the order of multiplication affects the sign of the result.
The cross product of complex numbers is commonly used in fields such as physics, engineering, and mathematics. It is particularly useful in vector calculus and electromagnetism, where it is used to calculate the direction and magnitude of forces and fields.