Geometry with Trigonometry
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6 Cartesian coordinates; applications COMMENT. Hitherto we have confined ourselves to synthetic or pure geometrical arguments ai ...
82 Cartesian coordinates; applications Ch. 6 perpendicular fromZtoOJ.Welet x= { |O,U| if Z∈H 3 , −|O,U| if Z∈H 4 , and y= { |O,V ...
Sec. 6.1 Frame of reference, Cartesian coordinates 83 Proof. (i) This is clear from the definition of coordinates. (ii) For ifZ ...
84 Cartesian coordinates; applications Ch. 6 and so|Z 1 ,Z 3 |=^12 |Z 1 ,Z 2 |. Similarly |Z 3 ,Z 2 |^2 = [ x 1 +x 2 2 −x 2 ] 2 ...
Sec. 6.2 Algebraic note on linear equations 85 6.2 Algebraicnoteonlinearequations 6.2.1 It is convenient to note here some resul ...
86 Cartesian coordinates; applications Ch. 6 6.3 Cartesianequationofaline 6.3.1 Given any line l∈Λ, there are numbers a,b and c, ...
Sec. 6.3 Cartesian equation of a line 87 Note thatb=0, asb=0wouldimplya=0 here. On inserting these values foraand cabove we see ...
88 Cartesian coordinates; applications Ch. 6 for some j= 0. Proof. Necessity. Suppose first thatlcan be expressed in each of th ...
Sec. 6.4 Parametric equations of a line 89 6.4 Parametric equations of a line 6.4.1 Let l be a line with Cartesian equation ax+b ...
90 Cartesian coordinates; applications Ch. 6 Ifb>0, thenx 0 <x 0 +bandsoby6.1.1U 0 ≤mU 1. In this case we say that the cor ...
Sec. 6.4 Parametric equations of a line 91 withxandyas in (6.4.2), by 6.1.1 we have |Z 0 ,Z|= √ (x−x 0 )^2 +(y−y 0 )^2 = √ (bt)^ ...
92 Cartesian coordinates; applications Ch. 6 (iv) [Z 0 ,Z 1 ={Z≡(x,y):x=x 0 +t(x 1 −x 0 ),y=y 0 +t(y 1 −y 0 ),t≥ 0 }. Proof. By ...
Sec. 6.6 Projection and axial symmetry 93 Asab 1 −a 1 b=0, by 6.2.1 these will have a unique solution, yielding a point which w ...
94 Cartesian coordinates; applications Ch. 6 (iii) sl(Z 0 )≡ ( x 0 − 2 a a^2 +b^2 (ax 0 +by 0 +c),y 0 − 2 b a^2 +b^2 (ax 0 +by 0 ...
Sec. 6.7 Coordinate treatment of harmonic ranges 95 6.6.3 Inequalities for closed half-planes................... Let l≡ax+by+c= ...
96 Cartesian coordinates; applications Ch. 6 ThusZdivides(Z 1 ,Z 2 )in the ratio|λ|:1. Changing our notation slightly, if we den ...
Sec. 6.7 Coordinate treatment of harmonic ranges 97 In (c)−^12 =t−^12 ort−^12 =^12 , so eithert=0ort=1. Similarly eithers=0or s= ...
98 Cartesian coordinates; applications Ch. 6 then x 1 = 1 1 +μ x 3 + μ 1 +μ x 4 ,y 1 = 1 1 +μ y 3 + μ 1 +μ y 4. If we defineμ′by ...
Sec. 6.7 Coordinate treatment of harmonic ranges 99 x 4 =x 1 + λ λ− 1 (x 2 −x 1 ),y 4 =y 1 + λ λ− 1 (y 2 −y 1 ), Nowλ/(λ− 1 )> ...
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