colinearity

Colinearity is a geometric concept describing the arrangement of points on a single straight line. In Euclidean space, a set of points is colinear if there exists a line that contains all of them. Any two points are trivially colinear; three or more points may or may not be.

In Cartesian coordinates, three points (x1, y1), (x2, y2), (x3, y3) are colinear if the area of the triangle they form is zero, which can be checked by the determinant formula: x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2) = 0. Equivalently, the direction vectors P1P2 and P1P3 are linearly dependent, or the slopes between pairs are equal (when defined). In three dimensions, all points lie on a common line; the cross product of direction vectors is zero.

More generally, a set of points {P0, P1, ..., Pk} is colinear if there exists a fixed point P0 and a direction vector v such that every Pi can be written as P0 + ti v for some scalar ti. In higher-dimensional spaces, colinearity corresponds to the coordinates forming a matrix of rank 2 or less.

Applications and related concepts include geometry problem solving, computer graphics, and, in statistics, the related notion of multicollinearity, where predictor variables exhibit strong linear relationships that can affect regression analyses. Detection methods include determinant tests, slope comparisons, cross products, or rank checks, often with tolerance settings for near-colinear data.