1. Mar 21, 2012

### hoopsmax25

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. Mar 21, 2012

### 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?

3. Mar 21, 2012

### hoopsmax25

yeah i understand that. i think thats where i am stuck at. i dont know what the next step would be.

4. Mar 21, 2012

### 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?

5. Mar 21, 2012

### hoopsmax25

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

6. Mar 21, 2012

### HallsofIvy

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