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!