MATH 150 PRELIMINARY NOTES 2

CARTESIAN COORDINATES IN THE PLANE

Let us review the Cartesian coordinate system. The Cartesian (or rectangular) coordinate system is made up of two number lines that are perpendicular to each other at the point (0, 0) called the origin. (See figure 1) These two number lines divide the plane up into four quadrants, numbered from 1 to 4 in a counterclockwise rotation. Notice that the positive x-values are to the right of the origin, and the negative x-values are to the left. Positive y-values are above the origin, and negative ones below. Any point in the plane can be represented by an ordered pair (x, y).

figure 1

 

DEFINITION:

An increment in a variable is a net change in that variable. If x changes from x 1 to x 2, then the increment in x is D x = x 2 - x 1.

 

EXAMPLE 1:

Find the coordinate increments for x and y when we move from C (-2, 5) to D (3, 6).

 

SOLUTION: D x = x 2 - x 1 = 3 - (-2) = 5 and D y = y 2 - y 1 = 6 - 5 = 1

 

DEFINITION:

THE DISTANCE FORMULA IN THE PLANE

The distance between P (x 1, y 1) and Q (x 2, y 2) is

 

EXAMPLE 2:

Find the distance from C (-2, 5) to D (3, 6).

 

SOLUTION:

 

DEFINITION:

The midpoint between the points P (x 1, y 2) and Q (x 2, y 2) is

 

EXAMPLE 3:

Find the midpoint between C (-2, 5) and D (3, 6).

 

SOLUTION:

STRAIGHT LINES

Given two points P (x 1, y 1) and Q (x 2, y 2), we will call D x = x 2 - x 1 the run and D y = y 2 - y 1 the rise. Also, remember from geometry that any two points determine a unique line, so we can determine the equation of a line using only two points. Any non-vertical line in the plane has the property that the ratio

is the slope of the line.

 

FACT:

The slope of a vertical line is undefined because the change in x is 0, and you cannot divide by zero.

FACT:

The slope of a horizontal line is 0.

 

DEFINITION:

The angle of inclination of a line that crosses the x-axis is the smallest counterclockwise angle from the x-axis to the line. (See figure 2)

figure 2

 

FACT:

The angle of inclination of a horizontal line is 0o.

FACT:

The angle of inclination of a vertical line is 90o.

 

Let j (the lower case Greek letter phi) be the angle of inclination, so 0o £ j < 180o, in fact, the slope, m = tan j .

PARALLEL AND PERPENDICULAR LINES

FACT:

Lines that are parallel have equal angles of inclination (i.e. they have the same slopes).

FACT:

If two non-vertical lines L1 and L 2 are perpendicular, then their slopes m1 and m2 satisfy m1m 2 = -1 or m1 = -1/m2 or m2 = -1/m1.

EQUATIONS OF LINES

FACT:

The equation for a vertical line is x = a.

FACT:

The equation for a horizontal line is y = b.

 

DEFINITION:

The equation y - y 1 = m(x - x 1) is the point-slope equation of the line that passes through the point (x 1, y 1) and has slope m.

DEFINITION:

The equation y = mx + b is called the slope-intercept equation of the line with slope m and y-intercept b.

 

EXAMPLE 4:

Find the equation for a) the vertical line and b) the horizontal line that passes through the point (2, -4).

 

SOLUTION: The equation of a vertical line has the form x = a, so the equation for the vertical line that passes through (2, -4) is x = 2. The equation of a horizontal line has the form y = b, so the equation for the horizontal line that passes through (2, -4) is y = -4.

 

EXAMPLE 5:

Write an equation for the line that passes through the points (2, 5) and   (-3, 10).

 

SOLUTION: First find the slope of the line.

Now use the point-slope equation to find the equation of the line. I will use the point (2, 5), but you could use either point.

y - 5 = -1(x - 2) ® y - 5 = -x + 2 ® y = -x + 7

 

EXAMPLE 6:

Find the equation of the line through the point (6, 2) and perpendicular to the line 6x - 2y = 10.

 

SOLUTION: First I must put 6x - 2y = 10 into slope-intercept form. From this form, I will determine the slope of this line.

6x - 2y = 10 ® -2y = 10 - 6x ® y = 3x - 5

The slope of this line is 3, so the perpendicular slope will be -1/3. Now to find the equation of the line.

The topic of this set of preliminary notes is the Cartesian coordinate system and related topics. The related topics include increments, distance formula, midpoint, slope, and equations of lines. These topics should be familiar to you, but if they are not, please see me for help. Work through the examples that are provided in this set of notes, and use this set of notes as a study guide when doing the homework.

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