Perhaps the single most important mathematical device used in computer graphics and many engineering and physics applications is the vector. A vector is a geometric object of a sort, because, as we will soon see, it fits our notion of a displacement or motion. (We can think of displacement as a change in position. If we move a book from a shelf to a table, we have displaced it a specific distance and direction.) Vector methods offer a distinct advantage over traditional analytic geometry by minimizing our computational dependence on a specific coordinate system until the later stages of solving a problem. Vectors are direct descendants of complex numbers and are generalizations of hypercomplex numbers. Interpreting these numbers as directed line segments makes it easier for us to understand their properties and to apply them to practical problems.

Length and direction are the most important vector properties. Scalar multiplication of a vector (that is, multiplication by a constant), vector addition, and scalar and vector products of two vectors reveal more geometric subtleties. Representing straight lines and planes using vector equations adds to our understanding of these elements and gives us powerful tools for solving many geometry problems. Linear vector spaces and basis vectors provide rigor and a deeper insight into the subject of vectors. The history of vectors and vector geometry tells us much about how mathematics develops, as well as how gifted mathematicians established a new discipline.