Understanding the Relationship between Vector Dot and Cross Products

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SUMMARY

The discussion centers on the mathematical relationship between vector dot products and cross products, specifically the expression (kxH)·k = 0. The user initially struggles to understand why this expression equals zero when expanded. It is clarified that the cross product kxH produces a vector perpendicular to both k and H, leading to a dot product with k that results in zero. This relationship is a fundamental property of vector operations in electromagnetism.

PREREQUISITES
  • Understanding of vector operations, specifically dot and cross products.
  • Familiarity with electromagnetic theory and vector calculus.
  • Knowledge of the properties of perpendicular vectors.
  • Basic experience with mathematical notation and manipulation.
NEXT STEPS
  • Study the properties of vector cross products in three-dimensional space.
  • Explore the implications of the curl operation in vector fields.
  • Learn about the applications of dot products in physics, particularly in electromagnetism.
  • Investigate advanced vector calculus techniques, including divergence and gradient.
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Students and professionals in physics, particularly those focusing on electromagnetism, as well as mathematicians and engineers dealing with vector calculus and its applications.

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Hi, I was looking at an EM problem today and realized I wasn't sure why

(kxH)\dotk = 0

I tried writing it out explicitly and got (w 1,2,3 representing directions)

A1(A2*B3-A3*B2) - A2(A1*B3-A3*B1) + A3(A1*B2-A2*B1)

and I can't see why this should equal zero. This is troubling because the whole dot product of a cross product where one of the vectors is repeated seems to be a general result (of course AxA = 0, that is fine). In fact, I realized I have been using it fairly frequently and wasn't sure why it was so. Any help would be appreciated.
 
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Well wouldn't kxH give a vector that is perpendicular to both k and H? That vector dot product with k wouild give what then?
 
damn it, i knew it was something that simple. of course the result of the curl operation will be a vector quantity that is perpendicular to both k and H. Then the dot product of perpendicular vectors is zero. Thanks for kicking me in the head ;D
 

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