Geometry with Trigonometry
180 Complex coordinates; sensed angles; angles between lines Ch. 10 Now (10.11.7) is a circle whenc= 2 h=0andg=−f. The equation ...
Sec. 10.11 A case of Pascal’s theorem, 1640 181 thenP,Q,R,Sare the vertices of a parallelogram. 10.2 IfZ 1 ∼z 1 ,Z 2 ∼z 2 andZ 3 ...
182 Complex coordinates; sensed angles; angles between lines Ch. 10 10.6 LetA,B,Cbe non-collinear points and takeD∈BC,E∈CA,F∈ABs ...
Sec. 10.11 A case of Pascal’s theorem, 1640 183 10.15 If[Z 1 ,Z 2 ,Z 3 ,Z 4 ]is a parallelogram,Wis a point on the diagonal line ...
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11 Vector and complex-number methods 11.1 Equipollence.............................. 11.1.1 .................................... ...
186 Vector and complex-number methods Ch. 11 (x)If Z 1 =Z 2 and Z 3 ∈Z 1 Z 2 ,then(Z 1 ,Z 2 )↑(Z 3 ,Z 4 )if and only if [Z 1 , ...
Sec. 11.2 Sum of couples, multiplication of a couple by a scalar 187 Recall thatZ 1 ,Z 2 have parameters 0 and 1 and 0<1. As ...
188 Vector and complex-number methods Ch. 11 Definition.ForO∈Π,letV(Π;O)be the set of all couples(O,Z)forZ∈Π.We define thesum(O, ...
Sec. 11.2 Sum of couples, multiplication of a couple by a scalar 189 (v) For alla,b∈V,a+b=b+a. (vi) Next,R×V→V is a function. (v ...
190 Vector and complex-number methods Ch. 11 11.3 Scalar or dot products ....................... 11.3.1 ........................ ...
Sec. 11.3 Scalar or dot products 191 But by 6.6.1(ii), πl(Z 2 )≡ ( x 2 + y 1 y^21 +x^21 (−y 1 x 2 +x 1 y 2 ),y 2 − x 1 y^21 +x^2 ...
192 Vector and complex-number methods Ch. 11 COMMENT. Now that we have set up our couples we call(O,Z)aposition vector with resp ...
Sec. 11.4 Components of a vector 193 11.4.2Arealcoordinates ........................... Given non-collinear pointsZ 1 ,Z 2 ,Z 3 ...
194 Vector and complex-number methods Ch. 11 and so for any pointZ 0 ≡F(x 0 ,y 0 ), x−x 0 =p(x 1 −x 0 )+q(x 2 −x 0 )+r(x 3 −x 0 ...
Sec. 11.5 Vector methods in geometry 195 wherep+q+r=1, can be deduced from this. For δF(Z 1 ,Z 2 ,pZ 4 +qZ 5 +rZ 6 ) =δF ( Z 1 , ...
196 Vector and complex-number methods Ch. 11 As the coefficients on each side add to 1, by the uniqueness in 11.4.2 we can equat ...
Sec. 11.5 Vector methods in geometry 197 From this s 1 −t = 1 −v 1 −w , t 1 −r = 1 −w 1 −u , r 1 −s = 1 −u 1 −v , and so by mult ...
198 Vector and complex-number methods Ch. 11 andsohave Z 4 = st st+( 1 −s)( 1 −t) Z 2 + ( 1 −s)( 1 −t) st+( 1 −s)( 1 −t) Z 3. Wi ...
Sec. 11.5 Vector methods in geometry 199 11.5.3 Desargues’ perspective theorem, 1648 A.D. ........... Let(Z 1 ,Z 2 ,Z 3 )and(Z 4 ...
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