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By Edgar Asplund; Lutz Bungart

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The sum of n vectors AiB1 (i = 1, 2, , n) is given by Z(bi-a;) =nb*-na* = n(b* - a*) where A* and B* are the mean centers of the initial points Ai and the terminal points Bi, respectively; hence 2;A;Bi = n A*B*. In particular, -4 -A1B1 + A2B2 = 2 A*B*, where A*, B* are the mid-points of A1A2 and B1B2, respectively. 10. Barycentric Coordinates. If P is any point in the plane of -3-3 --3 the reference triangle ABC, the vectors AP, BP, CP are linearly dependent (§ 5) ; hence «(p - a) + /3(p - b) + y(p - c) = 0, p=as+9b+-ic (1) «+6 + y The denominator is not zero; for, if a+a+y=0, then aa+(3b+yc=0, and A, B, C would be collinear, contrary to hypothesis.

24. Components of a Vector. We now supplement the notation of § 12 by writing the components of a vector u ui or ui according as the basis is ei or et. The components ui are called contravariant, the components ui covariant, for reasons given in § 148. Any vector now may be written in two forms: u = ulel + u2e2 + u3e3. u = ulel + u2e2 + u3e3i From (1) and (2), we obtain equations of the type u - e' = ul, (1) (2) u - el = ul ; all six are included in ui = u - ei ui = u ei, (3) (4) (i = 1, 2, 3).

Example 2. iterpretation. Thus (Fig. 15a) gives cdcosw=a2-b2, and, if we write c = a + b, d = a - b, w = anL,Ie (c, d). If a = b, c d = 0; then PQRS is a rhombus and the angle PRT may be inscribed in a semicircle about Q. We thus have two geometric theorems: 1. The diagonals of a rhombus cut at right angles. 2. An angle inscribed in a semicircle is a right angle. Moreover, from (a-b) (a - b) = we have the cosine law: d2 = a2+0 - 2ab cos 0. P a Q Fro. 15a VECTOR ALGEBRA 32 §15 Example 3. We also may interpret identities involving scalar products by regarding the vectors a, b, as position vectors OA, OB, from an arbitrary origin O.

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