Coplanarity

Coplanarity is concept that applies to vectors in R3\displaystyle \mathbb{R}^{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 u\displaystyle \vec{u}, v\displaystyle \vec{v}, and w\displaystyle \vec{w} are coplanar if and only if there exists real numbers a\displaystyle a and b\displaystyle b such that

u=av+bw\displaystyle \vec{u} = a \vec{v} + b \vec{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\displaystyle a or b\displaystyle 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\displaystyle a and b\displaystyle b that you just found.
  6. If LS=RS\displaystyle \text{LS} = \text{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]\displaystyle \left [ - 1 , 2 , 3 \right ], [4,1,2]\displaystyle \left [ 4 , 1 , - 2 \right ], and [14,1,16]\displaystyle \left [ - 14 , - 1 , 16 \right ] coplanar?

First, we write the linear combination equation:

[14,1,16]=a[1,2,3]+b[4,1,2]\displaystyle \left [ - 14 , - 1 , 16 \right ] = a \left [ - 1 , 2 , 3 \right ] + b \left [ 4 , 1 , - 2 \right ].

That gives us the three equations

14=a+4b,1=2a+b,16=3a2b\displaystyle - 14 = - a + 4 b , \quad - 1 = 2 a + b , \quad 16 = 3 a - 2 b.

Solving the system defined by the first two equations tells us that

a=109andb=299\displaystyle a = \frac{10}{9} \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad b = - \frac{29}{9}.

If we substitute those values into the third equation, we get

LS=16andRS=889\displaystyle \text{LS} = 16 \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad \text{RS} = \frac{88}{9}.

Since LSRS\displaystyle \text{LS} \ne \text{RS}, the system is inconsistent and therefore the vectors [1,2,3]\displaystyle \left [ - 1 , 2 , 3 \right ], [4,1,2]\displaystyle \left [ 4 , 1 , - 2 \right ], and [14,1,16]\displaystyle \left [ - 14 , - 1 , 16 \right ] are non-coplanar.