By Melvin Hausner
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Extra info for A Vector Space Approach to Geometry
In Fig. 12, find the ratios AG/GE and AF/FC. We should like A and B to have mass 2 and mass 3, respectively. 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.
Intuitively, we wish to have P → Q and P′ → Q′ determine the same vector if we can push P → Q into P′ → Q′ (no rotations allowed). Mathematically, we wish to identify P → Q and P′ → Q′ under certain conditions. These conditions are: (a) PQ || P′Q′. (b) Length of = length of . (c) Orientation of P → Q = orientation of P′ → Q′. Under these conditions we say that P → Q determines the vector and that . Thus (read: vector PQ) is the symbol used for the vector determined by P → Q and the equation is a shorthand way of writing all three conditions (a), (b), and (c), above.
K(mP + nQ) = kmP + knQ VI (Subtraction Law). If m > n, the equation mP = nQ + xX may be solved for the unknown mass-point xX. There is only one solution. 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.
A Vector Space Approach to Geometry by Melvin Hausner