Coplanarity is concept that applies to vectors in `RR^3`. A set of vectors is coplanar if they all lie on the same plane. Just as a single vector is always collinear with itself, a pair of vectors are always coplanar. With three vectors, sometimes they are and sometimes they aren’t.
Three vectors `vec u`, `vec v`, and `vec w` are coplanar if and only if there exists real numbers `a` and `b` such that
`vec u = avec v + bvec w`.
In other words, it must be possible to express each as a linear combination of the other two. To determine this, you should use the following method:
Are the vectors `[-1,2,3]`, `[4,1,-2]`, and `[-14,-1,16]` coplanar?
First, we write the linear combination equation:
`[-14,-1,16] = a[-1,2,3] + b[4,1,-2]`.
That gives us the three equations
`-14=-a+4b quad and quad``-1=2a+b quad and quad 16=3a-2b`.
Solving the system defined by the first two equations tells us that
`a = 10/9 qquad and qquad b = -29/9`.
If we substitute those values into the third equation, we get
`"LS" = 16 qquad and "RS" = 88/9`.
Since `"LS"!="RS"`, the system is inconsistent and therefore the vectors `[-1,2,3]`, `[4,1,-2]`, and `[-14,-1,16]` are non-coplanar.