Coplanar
Categories: Geometry stubs | Geometry
A set of points is said to be coplanar if and only if they lie on the same geometric plane. Three points are always coplanar.
Other 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.
Properties
If three 3-dimensional vectors <math>\mathbf{a}, \mathbf{b} </math> and <math>\mathbf{c}</math> are coplanar, and <math>\mathbf{a}\cdot\mathbf{b} = 0</math>:
<math>(\mathbf{c}\cdot\mathbf{\hat a})\cdot\mathbf{\hat a}\ +\ (\mathbf{c}\cdot\mathbf{\hat b})\cdot\mathbf{\hat b} \ =\ \mathbf{\hat c}</math>
Or, the vector resolutes of <math>\mathbf{c}</math> on <math>\mathbf{a}</math> and <math>\mathbf{c}</math> on <math>\mathbf{b}</math> add to give the original <math>\mathbf{c}</math>.