In this terminology, the adjective affine refers to everything involving lines and planes, but excludes distances, angles, and hence orthogonality. These latter notions are described as Euclidean, after the famous Greek mathematician. The former are perfectly modelled by vector spaces. To handle the latter, a number called the "dot product" must be associated with each pair of vectors. How is this done?
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Dot product and norm --------------------
The concept of the dot product was brought to light by William Hamilton in his work on quaternions, four-dimensional numbers that he introduced in 1843. Soon afterwards, mathematicians from the British school of mathematical physics, including James Clerk Maxwell, Oliver Heaviside, and Josiah Willard Gibbs, preferred to define it using coordinates. Let us work in the three-dimensional space of triples of real numbers. Let u = (x, y, z) and v = (x', y', z') be two such triples. Their dot product is defined by (u | v) = xx' + yy' + zz'. A straightforward calculation shows that (u | v) = (v | u), that (u1+u2|v) = (u1| v) + (u2| v), and that, for every real number λ, (λu | v) = λ(u | v). The first property is called symmetry, and the other two together constitute linearity in the first variable (linearity in the second variable follows immediately from symmetry). We can, of course, calculate the dot product of a vector with itself, obtaining (u | u) = x2 + y2 + z2. This quantity is always nonnegative, and is strictly positive whenever the vector u is nonzero. We even recognize the formula for the square of the distance. The square root of this quantity is therefore called the norm of u; it is denoted by N(u) or || u ||.
This approach is not entirely satisfactory, however, since it is defined for triples of scalars rather than for an arbitrary vector space E. If E is finite-dimensional, we could choose a basis B and reduce the problem to the case above, but B would then play a special role; what would happen if we changed it? Furthermore, this excludes infinite-dimensional vector spaces, which, as we shall see (in the box), can also be equipped with such structures. We therefore define a map f that assigns to two vectors u and v a real number f (u, v), with the properties of linearity in the first variable and symmetry, and satisfying f (u, u) > 0 whenever u is nonzero. The scalar f (u, v), denoted by (u | v) or, more simply, u.v, is called the dot product of the vectors u and v.