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 .Let
Z 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 }.