By Melvin Hausner
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Extra resources for A Vector Space Approach to Geometry
This concludes the statement of the axioms. In our interpretation, the closure law (I) merely states that two mass-points have a single center of mass. The commutative law (II) says that the center of mass depends on the mass-points and not on the order in which they are mentioned. The associative law (III) is the formal statement of our assumption that, in computing the center of mass, we may replace any subsystem with “its equivalent,” namely, the total mass concentrated at the center of mass.
The reader should draw a picture, and pay special attention to the distinction between AP/AB and PA/AB. This technique, while not overwhelmingly useful,* may be justified on familiar grounds: it works. Thus it may be justified merely as a memory device. But we shall see in Chapter 2 that there is more to it than this. Exercises In the following exercises, no proofs are required—the correct answers will do fine. 1. 3) if the lines through the vertices pass through a common point outside a triangle.
7. A heavy wire is bent in the form of a triangle. Where is its center of mass? ) 8. Problems such as Exercises 1 and 2 above often distress students because the given information does not serve to determine the figure. Comment on this. 4 FURTHER PHYSICAL MOTIVATIONS The methods of the previous section may also be applied to solid geometry. Furthermore, even in the plane, some algebra (rather than the arithmetic of the last section) is often convenient. As in the previous section, we illustrate the techniques by examples.
A Vector Space Approach to Geometry by Melvin Hausner