Coplanarity

Coplanarity is concept that applies to vectors in ${ℝ}^3\text{.}$ 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 $\overrightarrow{u}\text{,}$ $\overrightarrow{v}\text{,}$ and $\overrightarrow{w}$ are coplanar if and only if there exists real numbers $a$ and $b$ such that

$\overrightarrow{u}=a\overrightarrow{v}+b\overrightarrow{w}\text{.}$

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 $b\text{.}$
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 $\text{LS}=\text{RS}\text{,}$ 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 $[-\mathrm{1,2,3}]\text{,}$ $[\mathrm{4,1},-2]\text{,}$ and $[-14,-\mathrm{1,16}]$ coplanar?

First, we write the linear combination equation:

$[-14,-\mathrm{1,16}]=a[-\mathrm{1,2,3}]+b[\mathrm{4,1},-2]\text{.}$

That gives us the three equations

$-14=-a+4b\text{,}$ $-1=2a+b\text{,}$ $16=3a-2b\text{.}$

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

$a=\frac{10}{9}$ and $b=-\frac{29}{9}\text{.}$

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

$\text{LS}=16$ and $\text{RS}=\frac{88}{9}\text{.}$

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