By Melvin Hausner

The results of geometry and linear algebra on one another obtain shut consciousness during this exam of geometry’s correlation with different branches of math and technological know-how. In-depth discussions contain a evaluate of systematic geometric motivations in vector area concept and matrix idea; using the guts of mass in geometry, with an creation to barycentric coordinates; axiomatic improvement of determinants in a bankruptcy facing region and quantity; and a cautious attention of the particle challenge. 1965 edition.

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Example text

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.

7). This is so because vector algebra is more “complete” and familiar. Two examples will illustrate the meaning of this statement. First, in general, it was impossible to subtract mass-points. Thus the unhappy notion of a “negative mass-point” was needed if a larger mass-point was subtracted from a smaller one. Even then, the equation 3A = 3B + xX had no sensible mass-point solution xX. For our second example, we merely note that an equation such as 3X = 2A + 2B made no sense at all. Masses on both sides of an equation had to agree.

7. In terms of barycentric coordinates, state when a point P is interior to, on the boundary of, on an edge of, or is a vertex of the tetrahedron ABCD. 9 SOME ALGEBRAIC ANTICIPATION Thus far we have been somewhat cavalier in our treatment of algebra. To some extent this is justified because this is not an algebra text. In addition, we shall find that the algebra is more readily justified by the methods of Chapter 2. Thus we continue with this cavalier treatment and merely promise that some of its justification will follow in Chapter 2.

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