Coplanarity is concept that applies to vectors in . 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 , , and are coplanar if and only if there exists real numbers and such that
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:
- Write the linear combination expression as shown above. It doesn’t matter which of the three vectors is alone on the left-hand side.
- Extract three linear equations from the vector equation.
- Choose two of the equations and use them to solve for or .
- 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.
- Perform an LS/RS verification of the as yet unused equation with the values of and that you just found.
- If , the system is consistent and the vectors are coplanar. If not, the system is inconsistent and the vectors are non-coplanar.
Are the vectors , , and coplanar?
First, we write the linear combination equation:
That gives us the three equations
Solving the system defined by the first two equations tells us that
If we substitute those values into the third equation, we get
Since , the system is inconsistent and therefore the vectors , , and are non-coplanar.