Coplanarity

In geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. For example, three distinct points are always coplanar; but four points in space are usually not coplanar.

Points can be shown to be coplanar by determining that the scalar product of a vector that is normal to the plane and a vector from any point on the plane to the point being tested is 0. To put this another way, if you have a set of points which you want to determine are coplanar, first construct a vector for each point to one of the other points (by using the distance formula, for example). Secondly, construct a vector which is perpendicular (normal) to the plane to test (for example, by computing the cross product of two of the vectors from the first step). Finally, compute the dot product (which is the same as the scalar product) of this vector with each of the vectors you created in the first step. If the result of each dot product is 0, then all the points are coplanar.

Distance geometry provides a solution to the problem of determining if a set of points is coplanar, knowing only the distances between them.

Properties
If three 3-dimensional vectors $$\mathbf{a}, \mathbf{b} $$ and $$\mathbf{c}$$ are coplanar, and $$\mathbf{a}\cdot\mathbf{b} = 0$$, then


 * $$(\mathbf{c}\cdot\mathbf{\hat a})\cdot\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\cdot\mathbf{\hat b} = \mathbf{\hat c}, $$

where $$\mathbf{\hat a}$$ denotes the unit vector in the direction of $$\mathbf{a}$$.

Or, the vector resolutes of $$\mathbf{c}$$ on $$\mathbf{a}$$ and $$\mathbf{c}$$ on $$\mathbf{b}$$ add to give the original $$\mathbf{c}$$.