1. 1.3 The Cartesian Coordinate System
MAT 125 – Applied Calculus
1.3 The Cartesian Coordinate System

2. Today’s Class
We will be learning the following concepts in Section 1.3:
The Cartesian Coordinate System
The Distance Formula
The Equation of a Circle
We will be learning the following concepts in Section 1.4:
Slope of a Line
Equations of Lines
1.3 The Cartesian Coordinate System
Dr. Erickson

3. The Cartesian Coordinate System
At the beginning of the chapter we saw a one-to-one correspondence between the set of real numbers and the points on a straight line (one dimensional space). – 4 3 2 1
1.3 The Cartesian Coordinate System
Dr. Erickson

4. The Cartesian Coordinate System
The Cartesian coordinate system extends this concept to a plane (two dimensional space) by adding a vertical axis. 4 3 2 1
– 1
– 2
– 3
– 4 –
1.3 The Cartesian Coordinate System
Dr. Erickson

5. The Cartesian Coordinate System
The horizontal line is called the x-axis, and the vertical line is called the y-axis. y 4 3 2 1
– 1
– 2
– 3
– 4 – x
1.3 The Cartesian Coordinate System
Dr. Erickson

6. The Cartesian Coordinate System
The point where these two lines intersect is called the origin. 4 3 2 1
– 1
– 2
– 3
– 4 – y x
Origin
1.3 The Cartesian Coordinate System
Dr. Erickson

7. The Cartesian Coordinate System
In the x-axis, positive numbers are to the right and negative numbers are to the left of the origin. 4 3 2 1
– 1
– 2
– 3
– 4 – y 4 3 2 1
– 1
– 2
– 3
– 4
Negative Direction
Positive Direction x
1.3 The Cartesian Coordinate System
Dr. Erickson

8. The Cartesian Coordinate System
In the y-axis, positive numbers are above and negative numbers are below the origin. 4 3 2 1
– 1
– 2
– 3
– 4 – 4 3 2 1
– 1
– 2
– 3
– 4 x y
Positive Direction
Negative Direction
1.3 The Cartesian Coordinate System
Dr. Erickson

9. The Cartesian Coordinate System
A point in the plane can now be represented uniquely in this coordinate system by an ordered pair of numbers (x, y).
(– 2, 4)
(– 1, – 2)
(4, 3) x y
(3,–1) – 4 3 2 1 4 3 2 1
– 1
– 2
– 3
– 4
1.3 The Cartesian Coordinate System
Dr. Erickson

10. The Cartesian Coordinate System
The axes divide the plane into four quadrants as shown below. 4 3 2 1
– 1
– 2
– 3
– 4 – y 4 3 2 1
– 1
– 2
– 3
– 4
Quadrant II
(–, +)
Quadrant I
(+, +) x
Quadrant III
(–, –)
Quadrant IV
(+, –)
1.3 The Cartesian Coordinate System
Dr. Erickson

11. The Distance Formula
The distance between any two points in the plane may be expressed in terms of their coordinates.
Distance formula
The distance d between two points P1(x1, y1) and P2(x2, y2) in the plane is given by
1.3 The Cartesian Coordinate System
Dr. Erickson

12. Example 1 Find the distance between the points (– 2, 1) and (10, 6).
1.3 The Cartesian Coordinate System
Dr. Erickson

13. Example 2
Let P(x, y) denote a point lying on the circle with radius r and center C(h, k). Find a relationship between x and y.
Solution:
By definition in a circle, the distance between P(x, y) and C(h, k) is r. With distance formula we get
Squaring both sides gives
C(h, k) h k r
P(x, y) y x
1.3 The Cartesian Coordinate System
Dr. Erickson

14. Equation of a Circle
An equation of a circle with center C(h, k) and radius r is given by
1.3 The Cartesian Coordinate System
Dr. Erickson

15. Example 3
Find an equation of the circle with radius 3 and center (–2, –4).
Find an equation of the circle centered at the origin and passing through the point (2, 3).
1.3 The Cartesian Coordinate System
Dr. Erickson

16. Example 4
1.3 The Cartesian Coordinate System
Dr. Erickson

17. Please continue on to Section 1.4
1.3 The Cartesian Coordinate System
Dr. Erickson