What Can We Infer About Vector w Given Its Relation to u and v?

[Nx2]

hi guys just want to clarify something...

If we are givin (vectors w u and v) w=au +bv what can we say about w
a) if u and v are independent?
b) if u an v are independent?

this is what i got so far.
a) w cannot be writtin as a scalar multiple of u and v and are therefore not coplanar.
b)w can be written as a scalar multiple of u and v and are therefor coplanar.

but i think there is more to it. I don't think my answer explains enough.

in a) can i say that w=0?... just trying to find more properties of w for a) and b)... any help would be appreciated. thnks.

- Tu

 

[HallsofIvy]

No, w is coplanar with u and v by virtue of being a linear combination of them. What is true is that if u and v are independent, then w is not a multiple of one or the other and is not "co-linear" with either. If u and v are dependent, then one is a multiple of the other, w is a multiple of each, and w points in the same direction as both u and v (which are "co-linear" since they are dependent).

 

[Guppy]

We should know that any two vectors in a N-Dimensional space where N>1 should be coplanar since Plane is 2-dimensional , that is, any of the 2 vectors should lie on the same plane.

 

[MiniTank]

We should know that any two vectors in a N-Dimensional space where N>1 should be coplanar since Plane is 2-dimensional , that is, any of the 2 vectors should lie on the same plane.

what? I am pretty sure that's wrong dude, what about two vectors in three space? your thinking is limited to 2 dimensions or 2-space..BTW Nx2, does that happen to be in the Harcourt mathematics 12 geometry and discrete math textbook?

 

[Data]

In order to define a plane (through the origin), you need two (linearly independent) vectors. So it doesn't make much sense to talk about two vectors being coplanar. Sure, they're always both in some plane together, though, specifically, the plane that they define (if they're independent)!

 

[HallsofIvy]

what? I am pretty sure that's wrong dude, what about two vectors in three space? your thinking is limited to 2 dimensions or 2-space..BTW Nx2, does that happen to be in the Harcourt mathematics 12 geometry and discrete math textbook?

No, Guppy is completely correct and you are completely wrong. Any two vectors define either a line (if one is a multiple of the other) or a plane (if one is NOT a multiple of the other).

 

[Hippo]

Questions of that form appear in numerous texts, MiniTank, though it would be odd if you were right.

 

[MiniTank]

No, Guppy is completely correct and you are completely wrong. Any two vectors define either a line (if one is a multiple of the other) or a plane (if one is NOT a multiple of the other).

ya your right. my mistake