Coplanarity

Coplanarity is the property of geometric objects lying in one plane. It can be tested through coordinates, vector relations, or geometric constraints.

Plane (geometry): A flat, two-dimensional surface extending without bound in three-dimensional space. Coplanarity asks whether several objects fit within one such surface.
Coplanar points: A finite set of points that all lie in one plane. Their displacement vectors from one point provide a direct test for a shared plane.
Coplanar lines: Lines that lie in a single plane, whether or not they intersect. The condition distinguishes planar line arrangements from skew lines in three-dimensional space.
Collinearity: The property of points lying on one straight line. Collinearity is a stronger condition than coplanarity for a set of points.
Affine subspace: A translated linear subspace, including points, lines, and planes. A plane is an affine subspace, giving coplanarity a precise algebraic setting.
Plane equation: An equation describing all points in a plane, often written ax + by + cz = d. Substituting each object's coordinates checks whether the same plane contains them.
Skew lines: Two nonparallel lines in three-dimensional space that do not intersect and are not coplanar. They are the canonical failure of a pair of lines to share a plane.
Concurrency (geometry): The property of lines or other geometric objects passing through one common point. Concurrent lines need not lie in one plane when there are three or more.
Linear independence: A set of vectors is linearly independent when no vector is a linear combination of the others. Independence tests whether displacement vectors span enough dimensions to rule out a shared plane.
Matrix rank: The dimension of the row space or column space of a matrix. The rank of displacement vectors reveals whether their span fits within a plane.

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