The geometric setting in which a line can be defined is usually an affine space (a collection of points) containing that line. The most familiar affine space, the one in which we live, has dimension 3. Nevertheless, the affine plane (of dimension 2) is often used, particularly for making maps or in education to make geometry easier to learn. Working in higher-dimensional spaces of dimension greater than 3 is also useful in certain contexts, the best-known example being the four-dimensional spacetime of general relativity.
Once we have chosen the setting containing the line (D) that interests us, we still need to represent the points of this affine space and define the conditions that points must satisfy to lie on (D).
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The Cartesian representation
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The points of an affine space are most often defined by their coordinates in a coordinate system (known as Cartesian) consisting of a point (the origin, usually denoted O) and a number of linearly independent vectors V1, V2, V3… that span the associated vector space (they are said to form a basis of that space). The number of vectors in this basis is the dimension of the space.
