10 Complex coordinates; sensed angles; angles between lines
COMMENT. In this chapter we utilise complex coordinates, develop sensed angles
and rotations, complete our formulae for axial symmetries and identify isometries
in terms of translations, rotations and axial symmetries. We go on to establish more
results on circles and consider a variant on the angles we have been dealing with.
10.1 Complexcoordinates .........................
10.1.1 .....................................
We now introduce the field of complex numbers(C,+,.)as an aid. This has an added
convenience when doing coordinate geometry. We recall that anyz∈Ccan be written
uniquely in the formz=x+ıy,wherex,y∈Randı^2 =−1. We use the notations
|z|=
√
x^2 +y^2 ,z ̄=x−ıyfor the modulus or absolute value, and complex conjugate,
respectively, ofz. As well as having the familiar properties for addition, subtraction,
multiplication and division (except division by 0), these have the further properties:
z=z,z 1 z 2 =z 1 z 2 ,∀z,z 1 ,z 2 ∈C;z=ziffz∈R;
|z 1 z 2 |=|z 1 ||z 2 |,|z|=|z|,∀z,z 1 ,z 2 ∈C;|z|=ziffz∈Randz≥0;zz=|z|^2 ,∀z∈C;1
z=
z
|z|^2∀z= 0.Definition.LetF=([O,I,[O,J)be a frame of reference forΠand for any point
Z∈Πwe recall the Cartesian coordinates(x,y)ofZrelative toF,Z≡F(x,y).If
z=x+ıy, we also writeZ∼Fz, and callzaCartesian complex coordinateof the
pointZrelative toF.WhenFcan be understood, we can relax our notation and
denote this byZ∼z.
Complex coordinates have the following properties:-(i)|z 2 −z 1 |=|Z 1 ,Z 2 |for all Z 1 ,Z 2.Geometry with Trigonometry
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