6 Cartesian coordinates; applications
COMMENT. Hitherto we have confined ourselves to synthetic or pure geometrical
arguments aided by a little algebra, and traditionally this is continued with. This is a
difficult process because of the scarcity of manipulations, operations and transforma-
tions to aid us. The main difficulties in synthetic proofs are locational, to show that
points are where the diagrams suggest they should be, and in making sure that all
possible cases are covered.
For ease and efficiency we now introduce coordinates, and hence thoroughgoing
algebraic methods. These not only enable us to deal with the concepts already intro-
duced but also to elaborate on them in an advantageous way.
In Chapter 6 we do the basic coordinate geometry of lines, segments, half-lines
and half-planes. The only use we make of angles here is to deal with perpendicularity.
6.1 FRAME OF REFERENCE, CARTESIAN COORDI-
NATES
6.1.1
Definition. A couple or ordered pairF=([O,I,[O,J)of half-lines such thatOI⊥
OJ, will be called aframe of referenceforΠ. With it, as standard notation, we
shall associate the pair of closed half-planesH 1 ,H 2 , with common edgeOI,and
withJ∈H 1 , and the pair of closed half-planesH 3 ,H 4 , with common edgeOJ,
and withI∈H 3. We refer toQ 1 =H 1 ∩H 3 ,Q 2 =H 1 ∩H 4 ,Q 3 =H 2 ∩H 4 and
Q 4 =H 2 ∩H 3 , respectively, as the first, second, thirdandfourth quadrantsof
F. We refer toOIandOJas theaxesand toOas theorigin.
Given any pointZinΠ, (rectangular) Cartesian coordinates forZare defined as
follows. LetUbe the foot of the perpendicular fromZtoOIandVthe foot of the
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