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8 Forgetting the Origin 23 Fig. 18 −→ A + AB = B −−−−−−− −→ − A(A + → v )=→ v. 2. To define the sec−→ → ond operation, consider a point A and a vector − v = CD. Constructing the paral−→ → lelogram (A, B, D, C) as in Fig. 18 we thus have − v = AB. The second property announced in the statement does not leave us any choice, we must define → A+− v = B. This not only takes care of the second property, but also of the third one which −→ → simply reduces to AB = − v. The first property is proved analogously: consider again the parallelogram −→ −→ (A, B, D, C), which yields AC = BD and thus, by the parallelogram rule for adding vectors, −→ −→ −→ −→ −→ AB + BD = AB + AC = AD.
17 Constructing further −→ −→ −→ −→ −→ −→ OZ = OX + OC = (OA + OB) + OC we must prove that −→ −→ −→ −→ −→ −→ OZ = OA + (OB + OC) = OA + OY . In other words, we must prove that (O, A, Z, Y ) is a parallelogram, knowing by assumption that (O, A, X, B), (O, B, Y, C) and (O, X, Z, C) are parallelograms. The assumptions imply −→ −→ −→ XZ = OC = BY , thus (B, X, Z, Y ) is a parallelogram as well. Therefore −→ −→ −→ Y Z = BX = OA and (O, A, Z, Y ) is a parallelogram as expected. Now we present the result which underlies the modern definition of affine space on an arbitrary field, as studied in the next chapter.
35 The hyperbolic paraboloid that is, two lines intersecting at the origin. Cutting by the plane x = 0 yields the “downward directed” parabola z = −by 2 in the (y, z)-plane, while cutting by y = 0 yields the “upward directed” parabola z = ax 2 in the (x, z)-plane. The surface thus has the “saddle” shape depicted in Fig. 35 and is called a hyperbolic paraboloid. • ax 2 = z. All sections by a plane y = d are parabolas. The sections by a plane z = d are empty for d < 0; for d > 0 we obtain ⎧ ⎪ ⎨x = ± d a ⎪ ⎩ z=d that is, the intersection of two parallel planes with a third one: two lines parallel to the y-axis.