Lines in the Plane
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Lines in the Plane |
Printed by: | Guest user |
Date: | Tuesday, October 22, 2024, 1:43 AM |
Description
Read this section and work through practice problems 1-9.
Lines in the Plane
The first graphs and functions you encountered in algebra were straight lines and their equations. These lines were easy to graph, and the equations were easy to evaluate and to solve. They described a variety of physical, biological
and business phenomena such as relating the distance d traveled to the rate r and time t of travel, and for converting the temperature in Fahrenheit degrees (F) to Celsius (C).
The
first part of calculus, differential calculus, will deal with the ideas and techniques and applications of tangent lines to the graphs of functions, so it is important that you understand the graphs and properties and
equations of straight
lines.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-1.2-Lines-in-the-Plane.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.
The Real Number Line
The real numbers (consisting of all integers, fractions, rational and irrational numbers) can be represented as a line, called the real number line (Fig. 1). Once we have selected a starting location, called the origin, a positive direction
(usually up or to the right), and unit of length, then every number can be located as a point on the number line. If we move from a point to point on the line (Fig. 2), then we will have moved an increment of . This increment is denoted by the symbol ( read "delta x" ).
The Greek capital letter delta, ∆, will appear often in the future and will represent the "change" in something. If b is larger than a, then we will have moved in the positive direction, and will be positive. If b is smaller than a, then
will be negative and we will have moved in the negative direction. Finally, if is zero, then and we did not move at all.
We can also use the ∆ notation and absolute values to write the distance that we have moved. On the number line, the distance from to is
dist(a,b) =
or simply, dist(a,b) = .
The midpoint of the segment from to is the point on the number line.
Example 1: Find the length and midpoint of the interval from to .
Solution: . The midpoint is at .
Practice 1: Find the length and midpoint of the interval from to .
(Note:
Solutions to Practice Problems are given at the end of each section, after the Problems).
The Cartesian Plane
A real number plane (Fig. 3) is determined by two perpendicular number lines, called the coordinate axes, which intersect at a point, called the origin of the plane or simply the origin. Each point in the plane can be described by an ordered pair of numbers which specify how far, and in which directions, we must move from the origin to reach the point .
The point can then be located in the plane by starting at the origin and moving units horizontally and then y units vertically. Similarly, each point in the plane can be labeled with the ordered pair
which directs us how to reach that point from the origin. In this book, a point in the plane will be labeled either with a name, say , or with an ordered pair , or with both . This coordinate system is called
the rectangular coordinate system or the Cartesian coordinate system after Rene Descartes, and the resulting plane is called the Cartesian Plane.
The coordinate axes divide the plane into four quadrants which are labeled quadrants I, II, III and IV as in Fig. 4 We will often call the horizontal axes the x-axis and the vertical axis the y-axis and
then refer to the plane as the xy-plane. This choice of and y as labels for the axes is simply a common choice, and we will sometimes prefer to use different labels and even different units of measure on the
two axes.
Increments and Distance Between Points In The Plane
If we move from a point to a point in the plane, then we will have two increments or changes to consider. The increment in the or horizontal direction is which is denoted
by . The increment in the or vertical direction is . These increments are shown in Fig. 5 . does not represent times , it represents the difference in
the coordinates: .
The distance between the points and is simply an application of the Pythagorean formula for right triangles, and
The midpoint of the line segment joining and is
Example 2: Find an equation describing the points which are equidistant from and . (Fig. 6)
Solution: The points must satisfy so
By squaring each side we get
Then
so and , a straight line. Every point on the line is equally distant from and .
Practice 2: Find an equation describing all
points equidistant from and .
A circle with radius and center at the point consists of all points which
are at a distance of from the center : the points which satisfy .
Example 3: Find the equation of a circle with radius and center
. (Fig. 7)
Solution: A circle is the set of points which are at a fixed distance from the center point , so this circle will be the set of points
which are at a distance of 4 units from the point . P will be on this circle if .
Using the distance formula and simplifying,
.
Practice 3: Find the equation of a circle with radius and center .
The Slope Between Points In The Plane
In one dimension on the number line, our only choice was to move in the positive direction (so the x–values were increasing) or in the negative direction. In two dimensions in the plane, we can move in infinitely many directions and a precise means
of describing direction is needed. The slope of the line segment joining to , is
In Fig. 8, the slope of a line measures how fast we rise or fall as we move from left to right along the line. It measures the rate of change of the y-coordinate with respect to changes in the x-coordinate. Most of our work will
occur in 2 dimensions, and slope will be a very useful concept which will appear often.
If and have the same coordinate, then and . The line from to is vertical and the slope is undefined because . If and
have the same y coordinate, then and , so the line is horizontal and the slope is (assuming ).
Practice 4: For and , find , and the slope of the line segment from to .
If the coordinates of or contain variables, then the slope
is still given by , but we will need to use algebra to evaluate and simplify .
Example 4: Find the slope of the line segment from to . (Fig. 9)
Solution: and so and so , and the slope is .
In this example, the value of is the constant 2 and does not depend on the value of .
Practice 5: Find the slope and midpoint of the line segment from to .
Example 5: Find the slope between the points
and for .
Solution: and so . and so and the slope is
In this example, the value of depends on the values of both and .
Practice 6: Find the slope between for .
In application problems it is important to read the information and the questions very carefully.
Including the units of measurement of the variables can help you avoid "silly" answers.
Example 6: In 1970 the population of Houston was 1,233,535 and in 1980 it was 1,595,138. Find the slope
of the line through the points (1970, 1233535) and (1980, 1595138).
Solution:
But 36,160.3 is just a number which may or may not have any meaning to you. If we include the units of measurement along with the numbers we will get a more meaningful result:
which says that during the decade from 1970 to 1980 the population of Houston grew at an average rate of 36,160 people per year.
If the x–unit is time in hours and the y-unit is distance in kilometers, then m is , so the units for are kilometers/hour ("kilometers per hour"), a measure of velocity, the rate of change of distance with respect to time. If the x-unit is the number of employees
at a bicycle factory and the y-unit is the number of bicycles manufactured, then is , and the units for are bicycles/employee ("bicycles per employee"),
a measure of the rate of production per employee.
Equations of Lines
Every line has the property that the slope of the segment between any two points on the line is the same, and this constant slope property of straight lines leads to ways of finding equations to represent nonvertical lines.
Point–Slope Equation
In calculus, we will usually know a point on the line and the slope of the line so the point–slope form will be the easiest to apply, and the other forms of equations for straight lines can be derived from the point–slope form.
If is a
nonvertical line through a known point with a known slope m (Fig. 10), then the equation of the line is
Point-Slope:
Example 7: Find the equation of the line through (2,–3) with slope 5.
Solution: The solution is simply a matter of knowing and
using the point–slope formula. and so . This equation simplifies to (Fig. 11).
The equation of a vertical line through a point is . The only points on the vertical line through the point have the same x–coordinate as
.
Two–Point and Slope–Intercept Equations
If two points and are on the line , then we can calculate the slope between them and use the first point and the point–slope equation to get the equation of :
Two Points: where
Once we have the slope , it does not matter whether we use or as the point. Either choice will give the same simplified equation
for the line.
It is common practice to rewrite the equation of the line in the form , the slope-intercept form of the line. The line has slope and crosses the y-axis at
the point ( 0, b ).
Practice 7: Use the definition of slope to calculate the slope of the line .
The point-slope and the two-point formulas are usually more useful for finding the equation of a line, but the slope-intercept form is usually the most useful form for an answer because it allows us to easily picture the graph of the line and to quickly
calculate y-values.
Angles Between Lines
The angle of inclination of a line with the x-axis is the smallest angle θ which the line makes with the positive x-axis as measured from the x-axis counterclockwise to the line (Fig. 12). Since the slope
and since opposite/adjacent, we have that .
The slope of the line is the tangent of the angle of inclination of the line.
Parallel and Perpendicular Lines
Two parallel lines and make equal angles with the x-axis so their angles of inclination will be equal (Fig. 13) and so will their slopes. Similarly, if their slopes and are equal, then the equations of the lines will always differ by a constant:
which is a constant so the lines will be parallel. These two ideas can be combined into a single statement:
Two nonvertical lines and with slopes and are parallel if and only if .
Practice 8: Find the equation of the line in Fig. 14 which contains the point (–2,3) and is parallel to the line .
If two lines are perpendicular and neither line is vertical, the situation is a bit more complicated (Fig. 15).
Assume and are two nonvertical lines that intersect at the origin (for simplicity) and that and are points away from the origin on and , respectively. Then the slopes of and will be and . The line connecting and forms the third side of the triangle
, and this will be a right triangle if and only if and are perpendicular. In particular, and are perpendicular if and only if the triangle satisfies
the Pythagorean theorem:
or
.
By squaring and simplifying, this last equation reduces to
.
We have just proved the following result:
Two nonvertical lines and with slopes and are perpendicular if and only if their slopes are negative reciprocals of each other:
Practice 9: Find the line which goes through the point (2,–5) and is perpendicular to the line .
Example 8: Find the distance (the shortest distance) from the point (1,8) to the
line : .
Solution: This is a sophisticated problem which requires several steps to solve.
First we need a picture of the problem (Fig. 16). We will find the line through the point (1,8) and perpendicular to . Then we will find the point where and intersect, and, finally, we will find the distance from
to (1,8).
(i) has slope 1/3 so has slope , and has the equation which simplifies to .
(ii) We can find the point of intersection of and by replacing the in the equation for with the from so . Then so
, so and intersect at .
(iii) Finally, the distance from to (1,8) is just the distance from the point (1,8) to the point which is .
Angle Formed by Intersecting Lines
If two lines which are not perpendicular intersect at a point and neither line is vertical, then we can use some geometry and trigonometry to determine the angles formed by the intersection of the lines (Fig. 17). Since θ2
is an exterior angle of the triangle ABC, θ2 is equal to the sum of the two opposite interior angles so and . Then, from trigonometry,
The inverse tangent of an angle is between and ( –90o and 90o) so always gives the smaller of the angles.
The larger angle is or 180o – θo.
The smaller angle θ formed by two nonperpendicular lines with slopes and is
.
Example 9: Find the point of intersection and the angle between and . (Fig. 18)
Solution: Solving the first equation for y and then substituting into the second equation, so . Putting this back into either equation, we get . Each of the lines is
in the slope–intercept form so it is easy to see that and . Then
Practice Answers
Squaring each side and simplifying, we eventually have .
Practice 3: The
point is on the circle when it is 5 units from the center so . Then so
Practice 8: The line
has slope so the slope of the parallel line is .
Using the form and the point ( –2, 3) on the line, we have
Practice 9: The line has slope so the slope of the perpendicular line is
.
Using the form and the point ( 2, –5) on the line, we have so and or .