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.