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cross product in complex multiplication | Reply
Sorry for my ignorance.
Can some one explain why the complex part is the cross product? Thx.

<quote>
z = a + bi
w = c + di
zw = (ac + bd) + (ad − bc)i
Behold! The real part is the dot product, and the complex part is the cross product.
</quote>
Re: cross product in complex multiplication (response to post by flair) | Reply
The cross procuct actually is something in three dimensions. The cross product of (x1, y1, z1) and (x2, y2, z2) is
(y1 * z2 - y2 * z1, z1 * y2 - z2 * y1, x1 * y2 - x2 * y1).
Now if we see our two vectors as three dimensional vectors with z = 0, the cross product will have x = 0 and y = 0 and z = x1 * y2 - x2 * y1.

But looking at the part you quoted, shouldn't that be
zw = (a + bi) * (c + di) = ac + adi + bci + bdii = (ac - bd) + (ad + bc)i?
So that the real part is the cross product and the imaginary part is the dot product, instead of the other way around?
Re: cross product in complex multiplication (response to post by dskloet) | Reply
Maybe your browser is not showing the overline style. (Or maybe you just didn't notice it.) But anyway we're using the conjugate of z, so I believe this (just like the article says) is correct:
<span style="text-decoration: overline">z</span>w = (a - bi) * (c + di) = ac + adi - bci - bdii = (ac + bd) + (ad - bc)i
Re: cross product in complex multiplication (response to post by saintila) | Reply
Oh, ok. I just went by the quoted text. I see an overline in your post but not in the original post.
Re: cross product in complex multiplication (response to post by dskloet) | Reply
Sorry, I just copied it and that's it, without the overline..
I don't know why :(
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