being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms, but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians), a line is stated to have certain properties which relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions (i.e., the Euclidean plane), two lines which do not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. On an Euclidean plane, a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines. In higher dimensions In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n-dimensional space n−1 first-degree equations in the n coordinate variables define a line under suitable conditions. In more general Euclidean space, Rn (and analogously in every other affine space), the line L passing through two different points a and b is the subset The direction of the line is from a (t = 0) to b (t = 1), or in other words, in the direction of the vector b − a. Different choices of a and b can yield the same line. Collinear points Three points are said to be collinear if they lie on the same line. Three points usually determine a plane, but in the case of three collinear points this does not happen. In affine coordinates, in n-dimensional space the points X = (x1, x2, ..., xn), Y = (y1, y2, ..., yn), and Z = (z1, z2, ..., zn) are collinear if the matrix has a rank less than 3. In particular, for three points in the plane (n = 2), the above matrix