By Melvin Hausner

ISBN-10: 0486404528

ISBN-13: 9780486404523

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

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To designate positive numbers (the “masses”). A mass-point (mass m located at point P) will continue to be designated by mP. * Suppose we have k mass-points m1P1, . . , mkPk. We have assumed that they uniquely determine a new mass-point mP, where m = m1 + · · · + mk and where P is their center of mass. We shall write mP = m1P1 + · · · + mkPk Thus m1P1 + · · · + mkPk is a shorthand way of writing, “The mass-point obtained when all of the masses of m1P1, . . ” We have seen in the examples that the center of mass can be obtained by taking the centers of two points at a time, and repeating the operation.

In the tetrahedron of Fig. 37, determine G and H on BC and AD, respectively, so that GH passes through the mid-point of EF. 12. In the tetrahedron of Fig. 38, determine H so that GH meets EF. 39 13. In Fig. 39, determine conditions on the ratios a/b, . . , g/h in order that the lines EF and GH intersect. ABCD is a tetrahedron in space. 5 AN AXIOMATIC CHARACTERIZATION OF CENTER OF MASS We now investigate and formalize some of the basic assumptions which we have been making about the center of mass.

5 suggests that it should be possible to say that P → Q and P′ → Q′ have the “same orientation” even if is not parallel to . Discuss this. Recall that it is desirable to have reflexivity, symmetry, and transitivity. 5 7. Prove: If , then . Note that various degenerate cases may occur, and at least take note of them. 8. Rephrase Exercise 7 in more geometric terms.

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A Vector Space Approach to Geometry by Melvin Hausner

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