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Post by mondo on Aug 21, 2022 20:29:05 GMT
I would expect a lot more than "The conventional definition of $A \times B$ is intrinsically three-dimensional and it therefore presents a problem... $a \times b$ does not lie in the complex plane$" from a book that emphasis visual/geometrical understanding  So, I am looking for a supplement text or resource that covers why in essence the cross product gives us a perpendicular vector to the plane that contains both $a$ and $b$. Thank you. |
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Post by Admin on Aug 21, 2022 20:40:21 GMT
I would expect a lot more than "The conventional definition of $A \times B$ is intrinsically three-dimensional and it therefore presents a problem... $a \times b$ does not lie in the complex plane$" from a book that emphasis visual/geometrical understanding So, I am looking for a supplement text or resource that covers why in essence the cross product gives us a perpendicular vector to the plane that contains both $a$ and $b$. Thank you. Mondo It's just the way the cross product is defined. Any book on elementary vector analysis will discuss it. Needham redefines it for use in two dimensions (complex plane). It seems to me that Needham's book explains it all very geometrically on pages 27-30 in subsection 3 Vectorial Operations. Vasco |
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Post by mondo on Aug 23, 2022 7:13:26 GMT
He does show the connection between operations on a complex plane and dot/cross product but the orthogonality of the new vector resulting from a cross product $a \times b$ is not presented nor commented. The reason may be, as you suggested that this is assumed by definition. However there are proofs from algebra that show this by matrix calculations. The other one, more geometrical is when we try to calculate a dot product of any vector in the plane with the cross product of two vectors from the same plane - it should always give $0$. However here we can't show it as in this book the cross product is no longer a vector.
Thank you.
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Post by Admin on Aug 23, 2022 16:31:44 GMT
Mondo He does show the connection between operations on a complex plane and dot/cross product but the orthogonality of the new vector resulting from a cross product $a \times b$ is not presented nor commented. It is presented in figure 20b on page 28 and commented in the first 3 or 4 paragraphs on the same page . This fact is very clear since the cross product of two vectors in the complex plane is a vector perpendicular to the complex plane and so the dot product of this vector with any vector in the plane must be zero since the angle between them is $\pi/2$ and the cosine of $\pi/2$ is zero. But this is anyway irrelevant in VCA because everything is in two dimensions. Vasco |
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