Prove that the following three vectors aren't coplaner: {2,1,-1} and {-1,0,3} and {0,4,3}.

Expert Answers
mlehuzzah eNotes educator| Certified Educator

The short answer is to take the determinant of the of the matrix made up of these vectors.  If it is 0, they are coplanar.  If it is not 0, they are not coplanar.

  det `[[2,-1,0],[1,0,4],[-1,3,3]]`   = (2)(0)(3)+(-1)(4)(-1)+(0)(1)(3)-(0)(0)(-1)-(4)(3)(2)-(3)(-1)(1)=-17

The determinant is not zero, so the vectors are linearly independent, so they are not coplanar.


The longer explanation:

If the vectors are coplanar, then they are linearly dependent.  That is, (at least) one of them can be "built" by taking a combination of the other vectors.  

For example:

`2<2,1,-1>+3<-1,0,3> = <1,2,7>`

So the vectors `<2,1,-1>, <-1,0,3>, <1,2,7>`

are coplanar, since we can build `<1,2,7>`   out of a combination of  `<2,1,-1>`and `<-1,0,3>`

If the vectors are linearly independent, then each vector "adds another dimension" to the space.

So `{<2,1,-1>}`is a one-dimensional space,

`{<2,1,-1>,<-1,0,3>}` is a two-dimensional space (so all in one plane)

`{<2,1,-1>,<-1,0,3>,<0,4,3>}`  is a three-dimensional space (so not all in one plane, or not "coplanar")