Kolinearno

Kolinearno is a geometric property describing a set of points that all lie on a single straight line. In Euclidean space, a collection of points P1, P2, ..., Pk is kolinearno if there exists a line that contains every Pi. Equivalently, there is a fixed point A and a nonzero direction vector v such that each Pi can be written as Pi = A + ti v for some real parameter ti.

In two dimensions, three points P1(x1, y1), P2(x2, y2), P3(x3, y3) are kolinearno if and only if (x2 − x1)(y3 − y1) − (y2 − y1)(x3 − x1) = 0. This determinant condition is also expressed as the area of the triangle formed by the three points being zero, or by equal slopes between pairs of points.

In higher dimensions, kolinearno means that the vectors P1P2, P1P3, ..., P1Pk are all parallel to a common direction, i.e., they are linearly dependent. The concept generalizes from the familiar plane case: all points lie on one line, regardless of the ambient space’s dimension.

Two practical notes: any two distinct points are always kolinearno with each other, by definition. For three or more points, kolinearno is a special case of alignment; most randomly chosen points in a plane or space are not kolinearno.

See also: collinearity, linear geometry, vector methods.