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)^2 +(−at)^2 =|t|√
b^2 +a^2 ,|Z 1 ,Z|=√
(bt−t)^2 +(a−at)^2 =√
(t− 1 )^2 (b^2 +a^2 )=|t− 1 |√
b^2 +a^2 ,|Z 0 ,Z 1 |=√
b^2 +(−a)^2 =√
b^2 +a^2.Thus whent<0,|Z 0 ,Z|=(−t)√
b^2 +a^2 ,|Z 1 ,Z|=( 1 −t)√
b^2 +a^2 ,and so|Z,Z 0 |+|Z 0 ,Z 1 |=|Z,Z 1 |; thus by 3.1.2 and (i) above,Z 0 ∈[Z,Z 1 ],
Z 0 =Z,Z=Z 1.
When 0≤t≤1,
|Z 0 ,Z|=t√
b^2 +a^2 ,|Z,Z 1 |=( 1 −t)√
b^2 +a^2 ,and so|Z 0 ,Z|+|Z,Z 1 |=|Z 0 ,Z 1 |; thusZ∈[Z 0 ,Z 1 ].
Whent>1,
|Z 0 ,Z|=t√
b^2 +a^2 ,|Z 1 ,Z|=(t− 1 )√
b^2 +a^2 ,and so|Z 0 ,Z 1 |+|Z 1 ,Z|=|Z 0 ,Z|; thusZ 1 ∈[Z 0 ,Z]andZ=Z 0 ,Z=Z 1.
These combined show that the values oftfor which 0≤t≤1 are those for which
Z∈[Z 0 ,Z 1 ].
(iv) This follows directly from (ii) of the present theorem. It can also be proved as
follows. As in the proof of (iii) above, we see that the values oftfor whicht≥0are
those for whichZ∈[Z 0 ,Z 1.
COROLLARY.Let Z 0 ≡(x 0 ,y 0 )and Z 1 ≡(x 1 ,y 1 )be distinct points. Then the fol-
lowing hold:-
(i)Z 0 Z 1 ={Z≡(x,y):x=x 0 +t(x 1 −x 0 ),y=y 0 +t(y 1 −y 0 ),t∈R}.(ii)Let≤lbe the natural order on l=Z 0 Z 1 for which Z 0 ≤lZ 1 .LetZ 2 ≡(x 0 +t 2 (x 1 −x 0 ),y 0 +t 2 (y 1 −y 0 )),
Z 3 ≡(x 0 +t 3 (x 1 −x 0 ),y 0 +t 3 (y 1 −y 0 )).Then we have t 2 ≤t 3 if and only if Z 2 ≤lZ 3.(iii)[Z 0 ,Z 1 ]={Z≡(x,y):x=x 0 +t(x 1 −x 0 ),y=y 0 +t(y 1 −y 0 ), 0 ≤t≤ 1 }.