By Melvin Hausner

ISBN-10: 0486404528

ISBN-13: 9780486404523

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**Extra resources for A Vector Space Approach to Geometry**

**Sample text**

3, we certainly want . This suggests that PQ || P′Q′ [condition (a)] includes the case that line PQ is identical with line P′Q′. In fact, whenever we write AB || CD, we shall always mean that AB is parallel to CD in the usual sense, or that AB is the same line as CD. This usage of the symbol || or word parallel is in conformity with the notion that parallel lines are lines having the same direction. The exercises discuss this further. 4 shows PQ || P′Q′, in which the lengths are not equal. Yet it intuitively makes good sense to say that the orientations of P → Q and P′ → Q′ are the same.

This is the solution of the problem. 29 Exercises 1. In Fig. 30, determine masses at A, B, and C in such a way that G is the center of the mass of the resulting system. 2. In Fig. 31, find PT/TQ. 31 3. In Fig. 32, find BI/IC. 33 4. In Fig. 33, a, b, c, d, e, are given lengths. Find x. 5. In Fig. 34, C′ divides AB in the ratio 1/2, and similarly for A′ and B′. By assigning suitable masses to A, B, and C, and by suitably breaking them up, show that the centroid of A′B′C′ is also the centroid of ABC.

Clearly, any proportional numbers can be used. Since B and C need mass 5 and mass 2, we see that B “wants” to have mass 5 as well as 3. Choose 15, the least common multiple for the mass of B (in order to avoid fractions), and we quickly obtain Fig. 13, where we read In the exercises which follow, try to work as “physically” as possible, where this can be done. 13 Exercises 1. In Fig. 14, AP = 2PB and QC = 2PQ. Compare AR and RC. 15 2. In Fig. 15, find the ratios BP/PQ and AP/PR. 3. In Fig. 16, the letters a, b, c, d, and e represent actual distances.

### A Vector Space Approach to Geometry by Melvin Hausner

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