146 Complex coordinates; sensed angles; angles between lines Ch. 10
(ii)If Z 1 =Z 2 ,thenZ∈Z 1 Z 2 if and only if z−z 1 =t(z 2 −z 1 )for some t∈R.(iii)If Z 1 =Z 2 ,thenZ∈[Z 1 ,Z 2 if and only if z−z 1 =t(z 2 −z 1 )for some t≥ 0.(iv)If Z 1 =Z 2 ,thenZ∈[Z 1 ,Z 2 ]if and only if z−z 1 =t(z 2 −z 1 )for some t such
that 0 ≤t≤ 1.(v)For Z 1 =Z 2 and Z 3 =Z 4 ,Z 1 Z 2 ‖Z 3 Z 4 if and only if z 4 −z 3 =t(z 2 −z 1 )for
some t∈R\{ 0 }.(vi)For Z 1 =Z 2 and Z 3 =Z 4 ,Z 1 Z 2 ⊥Z 3 Z 4 if and only if z 4 −z 3 =tı(z 2 −z 1 )for
some t∈R\{ 0 }.Proof.
(i) For|z 2 −z 1 |^2 =|x 2 −x 1 +ı(y 2 −y 1 )|^2 =(x 2 −x 1 )^2 +(y 2 −y 1 )^2 =|Z 1 ,Z 2 |^2.
(ii) Forz−z 1 =t(z 2 −z 1 )if and only ifx−x 1 +ı(y−y 1 )=t[x 2 −x 1 +ı(y 2 −y 1 )].
If this happens for somet∈R,thenx−x 1 =t(x 2 −x 1 ),y−y 1 =t(y 2 −y 1 ). By 6.4.1,
Corollary (i), this implies thatZ∈Z 1 Z 2.
Conversely ifZ∈Z 1 Z 2 , by the same reference there is such at∈Rand it follows
thatz−z 1 =t(z 2 −z 1 ).
(iii) and (iv). In (ii) we haveZ∈[Z 1 ,Z 2 whent≥0 by 6.4.1, Corollary, and
similarlyZ∈[Z 1 ,Z 2 ]when 0≤t≤ 1.
(v) By 6.5.1, Corollary (ii),Z 1 Z 2 andZ 3 Z 4 are parallel only if
−(y 2 −y 1 )(x 4 −x 3 )+(y 4 −y 3 )(x 2 −x 1 )= 0. (10.1.1)We note that asZ 1 =Z 2 we must have eitherx 1 =x 2 ory 1 =y 2.
Suppose first thatz 4 −z 3 =t(z 2 −z 1 )for somet∈R.Then
x 4 −x 3 +ı(y 4 −y 3 )=t(x 2 −x 1 )+ıt(y 2 −y 1 ),andsoastis real,
x 4 −x 3 =t(x 2 −x 1 ),y 4 −y 3 =t(y 2 −y 1 ).Then
−(y 2 −y 1 )(x 4 −x 3 )+(y 4 −y 3 )(x 2 −x 1 )
=−(y 2 −y 1 )t(x 2 −x 1 )+t(y 2 −y 1 )(x 2 −x 1 )= 0 ,so that (10.1.1) holds and hence the lines are parallel.
Conversely suppose that the lines are parallel so that (10.1.1) holds. Whenx 1 =x 2 ,
we let
t=
x 4 −x 3
x 2 −x 1,
so thatx 4 −x 3 =t(x 2 −x 1 ). On inserting this in (10.1.1), we have
−t(y 2 −y 1 )(x 2 −x 1 )+(y 4 −y 3 )(x 2 −x 1 )= 0 ,