By Melvin Hausner
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Extra info for A Vector Space Approach to Geometry
In fact, whenever we write AB || CD, we shall always mean that AB is parallel to CD in the usual sense, or that AB is the same line as CD. This usage of the symbol || or word parallel is in conformity with the notion that parallel lines are lines having the same direction. The exercises discuss this further. 4 shows PQ || P′Q′, in which the lengths are not equal. Yet it intuitively makes good sense to say that the orientations of P → Q and P′ → Q′ are the same. We shall not at present go very deeply into this, although the notion is basic to item (c) in the definition of vector.
Finally, using Eq. (7), we obtain the answer to our problem: BE/EC = 5/4. The algebraically inclined will ask, “Why not solve for P as the unknown? ” This is done in the next example. 3 into an algebraic setting. 43 EXAMPLE 2. In Fig. 43, find P in terms of A, B, C and also find F in terms of A and C. Since we are in the land of algebra, we state the hypotheses in equation form and manipulate. We are given We want to find P in terms of A, B, and C, and also to find F (the intersection of BP and AC) in terms of A and C.
Thus coefficients may be equated in Eq. (24) when, and only when, A, B, C are not on the same line. 44 EXAMPLE 4. In Fig. 44, express P in terms of A, B, and C. * The left-hand side gives an arbitrary P on line EF. We break the right-hand side into a part containing B and C (this will be D), and a part containing A: Let Then Since P is supposed to be the mid-point of AD, we set the masses of A and D equal and obtain Substituting in (30) and using (31), we obtain Dividing by 2x, we have EXAMPLE 5.