Question

How do i know if the vectors are in the same plane of the space?

Answer 1

If you can write one of them as a linear combination of the other. A third vector will be at the same plane if it can be written as a linear combination of the two first vectors.

Answer 2

Thanks malu... For exemple: for the vectors u=(1,2,3) and v=(2,-3,-1) how write the linear combination?

Answer 3

Ohh! Sorry! Every 2 vectors define a plane! If they are linear combination of the other, they as paralell, have the same direction. These ones are non-paralel, because they cant be written as a linear combination. Just a third vector, if it is in tha same plane, is a linear combination from the first 2 vectors.

Answer 4

Then sorry i... If exist a third vector j= (7,4,5)?

Answer 5

Or better, in the case of v1=(1,0,-1) ; v2=(-1,0,1); v3=(2,-1,-1) ... This vectors lie in a plane?

Answer 6

The vectors v1, v2, v3 lie in a plane if the determinant is zero, ie., they have no volume in 3 space. Organize your vectors like so. Then calculate the determinant.

\[\left[\begin{matrix}v1_{1} & v1_{2} & v1_{3} \\ v2_{1} & v2_{2} & v2_{3} \\v3_{1} & v3_{2} & v3_{3} \end{matrix}\right]\]

Answer 7

av1 + bv2 = v3
(a,0,-a) + (-b,0,b) = (2,-1,-1)
a-b=2
0+0=-1, that cant happen, so they three are not in the same plane

Answer 8

Thank's again maludeleon and Waynex you both were very helpfully. Sorry for disturbing you two... Hugs from Brazil!