Geometry with Trigonometry

82 Cartesian coordinates; applications Ch. 6

perpendicular fromZtoOJ.Welet

x=

{

|O,U| if Z∈H 3 ,

−|O,U| if Z∈H 4 , and y=

{

|O,V| if Z∈H 1 ,

−|O,V| if Z∈H 2.

O I

J

H 1

H 2

H 4 H 3

Q 2 Q 1

Q 3 Q 4

Figure 6.1. Frame of reference.

O I

J H 1

H 2

H 4 H 3

Z

U

V

Figure 6.2. Cartesian coordinates.

Then the ordered pair(x,y)are calledCartesian coordinatesforZ, relative toF.We
denote this in symbols byZ≡F(x,y),butwhenFis fixed and can be understood,
we relax this notation toZ≡(x,y).

Cartesian coordinates have the following properties:-

(i)If Z∈Q 1 ,thenx≥ 0 ,y≥0;if Z∈Q 2 ,thenx≤ 0 ,y≥ 0 ;ifZ∈Q 3 ,then

x≤ 0 ,y≤ 0 ;ifZ∈Q 4 ,thenx≥ 0 ,y≤0.

(ii) If Z 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 )and

U 1 =πOI(Z 1 ),V 1 =πOJ(Z 1 ),U 2 =πOI(Z 2 ),V 2 =πOJ(Z 2 ),

then|U 1 ,U 2 |=±(x 2 −x 1 ),|V 1 ,V 2 |=±(y 2 −y 1 ).

(iii) If Z 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 ),then

|Z 1 ,Z 2 |=

√

(x 2 −x 1 )^2 +(y 2 −y 1 )^2.

(iv) If Z 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 )and Z 3 ≡(x 3 ,y 3 )where

x 3 =^12 (x 1 +x 2 ),y 3 =^12 (y 1 +y 2 ),

then Z 3 =mp(Z 1 ,Z 2 ).

(v)Let≤lbe the natural order on l=OI under which O≤lI. If x 1 <x 2 ,U 1 ≡

(x 1 , 0 )and U 2 ≡(x 2 , 0 ),thenU 1 ≤lU 2.