Vector addition - Vector Spaces

[depre87]

Homework Statement

Show if V is a vector space ([a,b,c]|ab>=0). I'm trying to test whether it is closed under
vector addition.

Homework Equations

v=[a1,b1,c1] w=[a2,b2,c2], v and w satisfy ab>=0
a1b1>=0, a2b2>=0
show (a1+a2)(b1+b2)>=0

The Attempt at a Solution

Got to a1b1 + a2b2 +a1b2 + a2b1 after expanding the above equation. Know a1b1 and a2b2 both >=0 however what do I do with the a1b2 and a2b1? I've attempted trying to factor them into a1b1 and have gotten a1b1[(b2/b1)+(a2/a1)] and know a1b1 >=0 however not sure what to do with the terms in the bracket to further prove whether or not it is closed under vector addition.

 

[micromass]

Hi depre87!

Have you tried searching a counterexample?

 

[depre87]

I just had a look and I don't understand it enough to apply it to my question, could you enlighten me? thanks for the suggestion though.

 

[micromass]

Just try some numerical values of v and w and see if you can come up with a counterexample...

 

[lanedance]

one way you could do it is to picture a & b in a 2D plane, the allowable areas are a&b both +ve, or both -ve. Now try and think of a vector addition that will take you outside of teh allowable areas (the a&b axes may be a good start)

 

[HallsofIvy]

Look at scalar multiplication rather than addition. If v is such a vector, what is (-1)v?

Or, similarly, every vector must have an additive inverse.