What is the identity for coplanar, non-collinear vectors?

[atavistic]

I kinda remember some identity which goes as follows:

If a,b,c are coplanar, non collinear vectors then

\alpha a + \beta b + \gamma c = 0
=> \alpha + \beta + \gamma = 0

or something like this. Can someone help me remember.

 

[radou]

Well, this isn't quite correct.

If a, b, and c are three coplanar vectors, they are for sure linearly dependent in the plane, since, if a, b are two non colinear non zero vectors in a plane, they form a basis, i.e. \alpha a + \beta b = 0 => \alpha = \beta = 0. Every third vector can be representet uniquely as a linear combination of the basis vectors a and b.

 

[mathman]

I kinda remember some identity which goes as follows:

If a,b,c are coplanar, non collinear vectors then

\alpha a + \beta b + \gamma c = 0
=> \alpha + \beta + \gamma = 0

or something like this. Can someone help me remember.

You need to replace "a, b, and c are coplanar" by "a, b, and c are NOT coplanar".

 

[rock.freak667]

Well, this isn't quite correct.

If a, b, and c are three coplanar vectors, they are for sure linearly dependent in the plane, since, if a, b are two non colinear non zero vectors in a plane, they form a basis, i.e. \alpha a + \beta b = 0 => \alpha = \beta = 0. Every third vector can be representet uniquely as a linear combination of the basis vectors a and b.

Also for coplanar vectors
a\cdot(b\times c)= 0

 

[atavistic]

I am sorry but I really mean \alpha + \beta + \gamma = 0 and not \alpha = \beta =\gamma = 0

 

[radou]

I am sorry but I really mean \alpha + \beta + \gamma = 0 and not \alpha = \beta =\gamma = 0

Well, if \alpha = \beta =\gamma = 0, then most certainly \alpha + \beta + \gamma = 0.

 

[HallsofIvy]

I kinda remember some identity which goes as follows:

If a,b,c are coplanar, non collinear vectors then

\alpha a + \beta b + \gamma c = 0
=> \alpha + \beta + \gamma = 0

or something like this. Can someone help me remember.

You need to replace "a, b, and c are coplanar" by "a, b, and c are NOT coplanar".

No, if they were not coplanar, that statement would not be true.

 

[mathman]

No, if they were not coplanar, that statement would not be true.

In his original statement he had all coef = 0, not the sum. Obviously changing the question would usually lead to a change in the response.

 

[HallsofIvy]

Oh, thanks. I hate it when people edit their post after there have been responses!