By Melvin Hausner
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Many of the tools defined during this e-book can be utilized with beauty changes to resolve move difficulties of better complexity. All makes an attempt were made to make the e-book self-contained.
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Extra resources for A Vector Space Approach to Geometry
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.
A Vector Space Approach to Geometry by Melvin Hausner