Coplanar
GeometryCoplanar points or lines are those that all lie within the same flat plane.
Definition
Coplanar means "in the same plane." Points or lines are coplanar if they all fit on the same flat surface - like all being on the same sheet of paper.
Example
All the corners of a rectangle are coplanar because they all lie on the same flat surface. But if you add a point floating above the paper, that point is NOT coplanar with the others.
Key Insight
Any three points are always coplanar (they fit on one flat surface). It takes four or more points to possibly be non-coplanar - like the four corners of a 3D box.
Definition
Points or lines are coplanar if there exists a single plane containing all of them. Three non-collinear points are always coplanar. Four or more points may or may not be coplanar. All points in a 2D figure (like a polygon) are automatically coplanar.
Example
The four vertices of a square are coplanar. The four vertices of a tetrahedron (triangular pyramid) are NOT coplanar - no single flat plane contains all four. Any three vertices of the tetrahedron, however, are coplanar.
Key Insight
Coplanarity matters in 3D geometry. Two lines in 3D that are coplanar either intersect or are parallel. Two lines that are NOT coplanar are called skew lines - they do not intersect and are not parallel.
Definition
Points $P_1, \ldots, P_n$ in $\mathbb{R}^3$ are coplanar if and only if the vectors $(P_2-P_1), (P_3-P_1), (P_4-P_1)$ are linearly dependent, i.e., the $3 \times 3$ determinant $\det([P_2-P_1, P_3-P_1, P_4-P_1]) = 0$. Equivalently, all points satisfy a single linear equation $ax+by+cz = d$.
Example
Four points are coplanar iff the scalar triple product $(P_2-P_1) \cdot [(P_3-P_1) \times (P_4-P_1)] = 0$. This determinant test extends to $n$ points: coplanar iff the rank of the matrix of displacement vectors is at most $2$.
Key Insight
Non-coplanar points define the simplest 3D figure, the tetrahedron. In linear algebra, coplanarity is equivalent to linear dependence of displacement vectors, linking geometry directly to the rank of a matrix.