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Proof about cross product

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that there is no vector ⃗e that has the property of the number 1 for cross product, namely
    that ⃗e × ⃗x = ⃗x for all ⃗x.

    2. Relevant equations

    I'm sorta stuck on how to show this.

    3. The attempt at a solution
    I set e=(e1,e2,e3) and x=(x1,x2,x3) and used cross product to multiply it out but got stuck there.
     
  2. jcsd
  3. Mar 21, 2012 #2

    jedishrfu

    Staff: Mentor

    cant you do this as a proof by contradiction using the definition of cross product?

    a x b = c (c is then perpendicular to a) and (c is perpendicular to b)

    so now you assume that e exists and then what do you get?
     
  4. Mar 21, 2012 #3
    yeah i understand that. i think thats where i am stuck at. i dont know what the next step would be.
     
  5. Mar 21, 2012 #4

    jedishrfu

    Staff: Mentor

    well if e exists then e x x = x right which means that x is perpendicular to e and x perpendicular to x bt can x be perpendicular to itself?
     
  6. Mar 21, 2012 #5
    Oh ok, so obviously it cannot. So is there a way to show that the contradiction by writing it out?
     
  7. Mar 21, 2012 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    One might also note that if [itex]\vec{x}[/itex] is any non-zero vector in the same direction as [itex]\vec{e}[/itex], then [itex]\vec{e}\times\vec{x}= \vec{0}\ne \vec{x}[/itex].
     
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