Understanding the Relationship between Vector Dot and Cross Products

[realcomfy]

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

(kxH)$$\dot$$k = 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.

 

[rock.freak667]

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?

 

[realcomfy]

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