Is there a vector that satisfies the property of the number 1 for cross product?

[hoopsmax25]

Homework Statement

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.

Homework Equations

I'm sort of stuck on how to show this.

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.

 

[jedishrfu]

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?

 

[hoopsmax25]

yeah i understand that. i think that's where i am stuck at. i don't know what the next step would be.

 

[jedishrfu]

yeah i understand that. i think that's where i am stuck at. i don't know what the next step would be.

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?

 

[hoopsmax25]

Oh ok, so obviously it cannot. So is there a way to show that the contradiction by writing it out?

 

[HallsofIvy]

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