How do tell if 3 vectors lie in a same plane?

If i have three vectors with A, B, C

how do you prove if all of them lie in a same plane?How do tell if 3 vectors lie in a same plane?Three vectors are in the same plane (coplanar) if and only if they are "dependent". That is, there is some linear combination Av1+ Bv2+ Cv3= 0 with not all of A, B, C equal to 0. Equivalently, one vector can be written as a linear combination of the other two.



let A = x1 i + y1 j + z1 k



B = x2 i + y2 j + z2 k



C = x3 i + y3 j + z3 k



when Determent of the matrix



...x1.....y1.....z1

...x2.....y2.....z2

...x3.....y3.....z3



is zero then the 3 vectors lie on same planeHow do tell if 3 vectors lie in a same plane?
If they all lie in the same plane, you will have 3 vectors in a two-dimensional subspace. This can only occur if they are linearly dependent. You can try to find a linear combination of one with respect to the other two, but the easiest way is to set them up as column vectors in a matrix, i.e.



[A|B|C] (I assume they are 3-vectors?)



and check the determinant. If the determinant is zero, the three are linearly dependent and thus must lie in the same plane. If the determinant is non-zero, they are linearly independent and do not lie in the same plane.How do tell if 3 vectors lie in a same plane?They will not be independent.



You prove that by showing that (for example)



A = kB + hC

where k and h are scalar coefficients (i.e., numbers, not vectors).



A plane is 2-dimensions and 2 vectors are sufficient to form the basis of a 2-dim space.



Let us say that you choose B and C as the basis, then any other vector (A or anything else on the same plane) can be formed by combining B and C.How do tell if 3 vectors lie in a same plane?
The cross produc of two vectors is perpendicular to the plane they occupy. So if you take cross products and get parallel vectors as the result, the original vectors are coplanar.How do tell if 3 vectors lie in a same plane?All previous answers look correct, but it is probably simplest to take (AxB)dotC=0 if coplanar. This is the same as det|ABC|=0.