What is cross product of two complex numbers?

[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.