## Coplanarity

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:

1. Write the linear combination expression as shown above. It doesn’t matter which of the three vectors is alone on the left-hand side.
2. Extract three linear equations from the vector equation.
3. Choose two of the equations and use them to solve for a or b.
4. Back-substitute the variable you just found into one of the two chosen equations, not into the unused one, and solve for the remaining variable.
5. Perform an LS/RS verification of the as yet unused equation with the values of a and b that you just found.
6. If "LS" = "RS", the system is consistent and the vectors are coplanar. If not, the system is inconsistent and the vectors are non-coplanar.

### Example

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.