3.1 prologue1 3 coordinates 3.1 prologue coordinates describe the location of a point or an object...

1

3 Coordinates

3.1 Prologue Coordinates describe the location of a point or an object using numbers. If you are shipwrecked somewhere off the coast of Patagonia and radioing for help, you can say, “I am somewhere off the coast of Patagonia.” Rescuers will arrive sooner, though, if you give a more precise location; for example, “I am 50 miles northeast of Cape Horn.” It would be even better to consult your GPS (Global Positioning System) device and report: “I am at 55.47235°S 66.18542°W.” A latitude-longitude pair serves as coordinates of a point on Earth. Coordinates connect geometry with algebra and analytical methods; computations replace measuring. When you move a cursor on a computer screen with a mouse, the computer converts the signals received from the mouse into numbers, computes the cursor coordinates on the screen, then displays the cursor at the appropriate location. Direct measuring of distances and angles is a slow, low-tech process. It may work if you want to mark a boundary around your field, lay out the foundation for a house, or measure the distance between two points on a map with a compass and a ruler. But if you need to launch a satellite into orbit, control an airplane on autopilot, or generate computer animations for special effects in a movie, you need some serious computations. Knowing the coordinates of points, on the plane and in space, we can use a computer to calculate distances and directions between them. It takes only one number to describe the location of a point on a line or a curve. For example, a milepost or an exit number may describe your location on a highway. The location of a point on the number line is described by the number it represents. It takes two numbers to describe a location on a plane or a surface and three numbers to describe a location in space. If you travel in time, it takes four numbers to describe your location in time and space. Lines are said to have one dimension; surfaces and planes, two dimensions; space, three dimensions; and space-time, four dimensions. Mathematicians and physicists often work with multi-dimensional spaces.

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2 CHAPTER 3 ~ COORDINATES

3.2 Cartesian Coordinates As we saw in Section <...>, there is a one-to-one correspondence between real numbers and points on the number line. Now let us consider two number lines on a plane, such that they are perpendicular to each other and intersect at zero. Let’s call them x and y axes. By convention, the x-axis is drawn horizontally, pointing to the right, and the y-axis is drawn vertically, pointing up:

y

x0 -1

-2

3

1

2

-3

1 2 3 -1 -2 -3

For any point P on the plane, we can draw a line through P perpendicular to the x-axis. Suppose that line intersects the x-axis at point M. Another line through P, perpendicular to the y-axis, intersects the y-axis at point N:

y

x0 -1

-2

3

1

2

-3

1 2 3 -1 -2 -3

P

M

N

The points M and N are called the projections of P to the x- and y-axis, respectively. The points M and N correspond to some numbers x and y on the respective number lines. The ordered pair (x, y) is called the Cartesian coordinates of the point P. (It is called an “ordered” pair because the order in which x and y are listed matters: the first number is the x-coordinate, the second the y-coordinate. We will write P(x, y) to indicate that x and y are the coordinates of P.

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CHAPTER 3 ~ COORDINATES 3

The x- and y-axes define a coordinate system on the plane. The intersection of the axes is called the origin of the coordinate system.

The coordinates of the origin are (0, 0). The axes divide the plane into four regions called quadrants (Figure 3-1). They are numbered in counterclockwise order.

Quadrant I Quadrant II

Quadrant III Quadrant IV

Figure 3-1. The x and y axes divide the plane into four quadrants

In the first quadrant 0x ≥ and 0y ≥ .

Example 1

A B

C

y

x0 -1

-2

3

1

2

-3

1 2 3 -1 -2 -3

What are the coordinates of the points A, B, and C above? Solution A(-3, 1), B(2, 2), and C(1, -2).

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Example 2 In what quadrant does the point P(6, -3) reside? Solution Since its x-coordinate is positive and its y-coordinate is negative, P resides in the fourth quadrant.

In a three-dimensional space, we need three numbers to describe the location of a point. In the Cartesian system, we take three axes, x, y, and z, intersecting at the origin, each perpendicular to the other two. For each point P we can take its projection to the “horizontal” plane x-y, and use the x-y coordinates of that projection as the x and y coordinates of P. The “vertical” position of P with respect to the x-y plane (positive or negative, depending on whether P is above or below the x-y plane), gives the third coordinate, z:

x

P y

z

Cartesian coordinates are named after René Descartes (1596-1650), the French philosopher, mathematician, and scientist who invented them (his Latin name was Renatus Cartesius).

A Cartesian coordinate system establishes a one-to-one correspondence between ordered pairs of real numbers and points on the plane. In general, in any coordinate system, the coordinates of a point describe the point’s location without ambiguity. But the converse is not required: different coordinates may describe the same location. For example, in the latitude-longitude coordinate system, the latitude 90°N corresponds to the North Pole, regardless of the longitude. Polar coordinates, explained in Section <...>, are another example: different coordinate pairs describe the same location on the plane.

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We can shift and/or rotate one or both axes in a Cartesian coordinate system and obtain a new Cartesian coordinate system. If we know what the transformation is, we can write simple equations that relate the new coordinates to the old ones. For example, if we shift the x-axis up by a positive distance b and the y-axis to the right by a positive distance a —

y

x

x

y

a

b

P

O

O

— the new coordinates ( , )x y of a point can be computed from its old coordinates ( , )x y as follows:

x x ay y b= −= −

These formulas work for negative a (shift to the left) and negative b (shift down), too.

Exercises 1. Look up René Descartes’ biography on the Internet. Which one of his books

introduced coordinates?

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2.

A

B

C

y

x 0 -1

-2

3

1

2

-3

1 2 3 -1 -2 -3

D Determine the coordinates of the points A, B, C, and D above. Which of these points is in the second quadrant?

3. Mark the points A(0, 3), B(-1, -2), and C(3, -1) on a Cartesian coordinate

grid. 4. We can describe the first quadrant as a set of all points for which

0, 0x y≥ ≥ . Describe the second, third, and fourth quadrants in a similar manner.

5. Name the quadrants in which the points (1, -2) , (-1, 2) , and (-1, -2) reside

without drawing the points. 6. Give an example of the coordinates of points O, A, B, and C such that O is

the origin, A is not on either axis, and OABC is a square.

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7. TrueType® scalable computer fonts represent each character by describing its contour. Each contour is split into line segments and quadratic splines. A spline is a segment of a smooth curve, which is inscribed into an angle whose sides are tangent to the contour:

Each spline is described by three points: the two points of tangency and the vertex of the angle. The following list of 25 points defines the contour of a character:

(-3, 7), (-3, 7), (-3.5, 7.5), (-1, 12), (3, 10), (8, 8), (3, 3), (9, 0), (4.5, -5), (1, -9), (-3, -7), (-4.5, -6.5), (-3.5, -5.5), (-3, -4.5) (-2.5, -5 ), (4, -9), (4, -3), (4, 1.5), (-1, 1.5), (-1, 1.5), (-1, 2), (5, 3), (3, 7), (1, 11), (-3, 7)

This contour consists of 12 segments. The first point, third point, and so on are on the contour; the points in between are the vertices of spline angles. (A duplicate point indicates that the spline is actually a straight line segment.) Sketch the character.

8. If we shift the x-axis 3 units to the right and the y-axis 2 units up, what are

the coordinates of P(4, -1) in the new coordinate system? 9. If we shift the x-axis 3 units to the left and the y-axis 2 units down, what are

the coordinates of P(4, -1) in the new coordinate system? 10. In computer graphics systems, the origin of the coordinate system is often

placed in the upper-left corner of the screen, with the x-axis pointing to the right and the y-axis pointing down. If the screen is 1680 by 1050 pixels (picture elements), what are the screen coordinates of a point located one quarter of the screen to the right and three quarters of the screen down?

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11. In computer graphics, a picture within a window on the screen is often described in coordinates relative to that window (with the origin in the upper-left corner of the window and the y-axis pointing down). That way, if you move the window on the screen, the description of the picture does not change. Computer software and/or hardware can translate the relative window coordinates into absolute screen coordinates. What are the absolute screen coordinates of a point, if its window coordinates (in pixels) are (100, 150), the size of the window is 400 by 450, and the window is centered on the 1680 by 1050 screen?

12. If we rotate the coordinate axes by 90° counterclockwise around the origin,

what are the coordinates of P(4, -1) and Q(3, 5) in the new coordinate system?

13. Given three points, (1, 7)A , (2, 2)B , and (6, 1)C , what should be the

coordinates of the point D for ABCD to be a parallelogram? 14. Propose a necessary and sufficient condition for the points 1 1( , )A x y ,

2 2( , )B x y , 3 3( , )C x y , and 4 4( , )D x y to form a parallelogram. 15. What are the coordinates of the midpoint of a straight line segment that

connects the points 1 1( , )A x y and 2 2( , )B x y ? 16. Prove algebraically (not using geometry) that the midpoints of the sides of

any quadrilateral form a parallelogram. Hint: see Questions 14 and 15.

3.3 Distance Recall that the distance between two points on a number line is equal to the absolute value of the difference of the corresponding numbers:

x1 0 1 2 3 -1 -2 -3

x2

2 1d x x= −

This is the length of the segment that connects the points. The same concept applies to the coordinate plane:

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The distance between two points on the plane is equal to the length of the line segment that connects the points.

If the points 1 1( , )A x y and 2 2( , )B x y both lie on the x-axis, or if AB is parallel to the x-axis, then the distance between A and B is equal to 2 1x x− . If A and B both lie

on the y-axis, or if AB is parallel to the y-axis, then the distance between A and B is equal to 2 1y y− . In the general case, we use the Pythagorean theorem to find the

distance between A and B (Figure 3-2): 2 222 1 2 1d x x y y= − + − .

2y

1y

1x 2x

2 1y y−

y

x 2 1x x−

d 1 1( , )A x y

2 2( , )B x y

Figure 3-2. 2 222 1 2 1d x x y y= − + −

This formula works for any positions of points A and B on the plane, regardless of the signs of 1 2 1 2, , ,x x y y and regardless of the relationships between these numbers, including the cases when AB is parallel to the x-axis (then 2 1y y= ), when it is parallel to the y-axis (then 2 1x x= ), even when A and B are the same point! Since

2 2c c= for any real number c, we can use parentheses instead of the absolute values: 2 2 2

2 1 2 1( ) ( )d x x y y= − + − , or

2 22 1 2 1( ) ( )d x x y y= − + − . This is called the distance formula.

Example 1 What is the distance between A(0, -2) and B(-1, 3)?

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Solution

( ) ( ) ( ) ( )2 2 2 2( 1) 0 3 ( 2) 1 5 1 25 26d = − − + − − = − + = + = .

The distance from the origin of a point ( , )P x y is 2 2r x y= + .

2 2 2r x y= + .

This is a special case of the distance formula, because the coordinates of the origin are (0, 0). In geometry we measured distances with a ruler; now we can use the distance formula to compute the distance between two points with known coordinates. In theory, we can now prove any geometry theorem and solve any geometry problem using only algebra!

Example 2 Find the height of an equilateral triangle if the length of its side is a. Solution Let us solve this problem without drawing a sketch, without using geometry at all. Let ABC be an equilateral triangle: AB BC AC a= = = . Let us introduce a coordinate system with the origin at C and the x-axis going along the line CA , so that the coordinates of C are (0, 0) and the coordinates of A are (a, 0). Suppose the coordinates of B are (x, y). Then y is the height of the triangle, so we need to find y . We have AB BC a= = . From the distance formula,

2 2 2 2BC a x y= = + 2 2 2 2( )AB a x a y= = − + Comparing the first equation to the second we get 2 2( )x x a= − . Recall that

2 2 2( ) 2x a x ax a− = − + . So 2 2 22x x ax a= − + ⇒ 22 22aax a x a x= ⇒ = ⇒ = .

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Plugging 2ax = into the first equation we get:

22 2

2aa y⎛ ⎞= + ⇒⎜ ⎟

⎝ ⎠

2 2 22 2 2 3

2 4 4a a ay a a⎛ ⎞= − = − = ⇒⎜ ⎟

⎝ ⎠ 3

2y a= .

The concept of distance is very general: in different contexts and in different mathematical theories, distance may mean different things. If you ask, for example, what is the distance from Atlanta to Boston, you probably do not mean the length of the line segment that connects Atlanta and Boston in three-dimensional space (unless you are going to build a tunnel from Atlanta to Boston). More likely, you are interested in the distance along the surface of the Earth, or, if you are driving, the travel distance along Interstate 95. On a chessboard, if you are only allowed to move horizontally and vertically, you might find it useful to consider the distance between the points 1 1( , )A x y and 2 2( , )B x y defined as 2 1 2 1( , )d A B x x y y= − + − , which is different from the standard distance formula. But no matter how you define the distance ( , )d A B , a well-behaved distance must always have the following three properties: ( , ) 0d A A = — the distance from any point to itself is 0. ( , ) ( , )d A B d B A= — the distance is symmetric: for any two points A and B,

the distance from A to B is equal to the distance from B to A. ( , ) ( , ) ( , )d A B d B C d A C+ ≥ — the triangle rule: the sum of the lengths of two

sides of a “triangle” never exceeds the length of the third side; that is, the “direct route” from A to C is the shortest.

For the distance formula on the coordinate plane, the first two properties are obvious. At first sight, the triangle rule is obvious, too: anyone knows that a straight line is the shortest distance between two points. But how do we “know” it, mathematically speaking? Precisely from the triangle rule! Its proof actually takes some work. The first step is to prove the triangle rule for the number line: for any real numbers

1 2 3, , andx x x , 2 1 3 2 3 1x x x x x x− + − ≥ − . Here the triangle is “squashed” into one dimension. The proof in this case is straightforward: we simply consider different configurations of 1 2 3, , andx x x . 2 1 3 2 3 1x x x x x x− + − = − when 2x is between 1x and 3x , and 2 1 3 2 3 1x x x x x x− + − > − when 2x is outside 1 3[ , ]x x .

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The second step is to prove the triangle rule on the plane for the special case when A, B, and C lie on the same straight line. The proof follows from the previous step: just turn the line into a number line. In the final step, we prove the general triangle rule. To do that, we use the projection of B onto the line AC . Let us denote that point as P :

A

B C

P

From the triangle rule for points on the same line, AP PC AC+ ≥ . But AP AB≤ and PC BC≤ . So, AB BC AC+ ≥ .

AB BC AC+ = if and only if the points A, B, and C lie on the same straight line, and B lies between A and C.

A set of points with a well-behaved distance defined for any two points is called a metric space.

Example 3 Let us consider the set of all possible positions of the minute hand on the face of a clock and define 1 2( , )d T T as the number of minutes it takes the minute hand to travel from the position 1T to the position 2T . Can we use 1 2( , )d T T as the definition of distance between 1T and 2T ? In other words, does 1 2( , )d T T have the three properties of a well-behaved distance? Solution It is easy to see that ( , ) 0d T T = , and that the triangle rule is satisfied, too. But

1 2( , )d T T is not symmetric: typically 1 2 2 1( , ) ( , )d T T d T T≠ . For example, it takes five minutes for the minute hand to travel from 2 to 3 and 55 minutes to travel from 3 to 2. Therefore, 1 2( , )d T T is not an acceptable definition for distance.

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Exercises 1. What is the distance on the coordinate plane between A(-4, 1) and

B(1, -2)? 2. What is the distance of P(-5, 12) from the origin? 3. The vertices of a triangle have the coordinates (-2, 2), (0, -2), and (1, 3). Is

this triangle equilateral, isosceles, or scalene? Find its perimeter. 4. The lengths of the bases of an isosceles trapezoid are 6 and 10; the distance

between the bases is 3. Draw this trapezoid in a convenient place on the coordinate plane and find the lengths of its side and the diagonals, using the distance formula.

5. Show, using the distance formula, that A(-4, -2), B(-1, -1), and C(5, 1) lie

on the same straight line. 6. Using coordinates, prove that for any rectangle ABCD and any point M on

the plane, inside or outside the rectangle, as well as on its border, 2 2 2 2MA MC MB MD+ = + . Hint: choose a convenient origin and

directions of the x- and y-axes. 7.

What is the length of the diagonal of a cube if the length of its side is 1 unit?

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8. The distance between two points in the three-dimensional space is defined as the length of the line segment that connects the points. Come up with a formula for the distance between 1 1 1( , , )A x y z and 2 2 2( , , )B x y z . Hint:

1 1( , , 0)x y x

B y

z

2 2( , , 0)x y

A 2 1z z−

9. Recall that a line segment that connects a vertex of a triangle with the

midpoint of the opposite side is called a median of the triangle. If 1 1( , )A x y ,

2 2( , )B x y , and 3 3( , )C x y are the vertices of a triangle, the point

1 2 3 1 2 3,3 3

x x x y y y+ + + +⎛ ⎞⎜ ⎟⎝ ⎠

is called the center of gravity of the triangle.

Using only the distance formula, show that the center of gravity of a triangle lies on each of its medians (and, therefore, all three medians intersect at the center of gravity). Hint: see Question <...> in Section <...>

10. The distance between two points on Earth’s surface is measured by the

length of the shortest arc that connects the points. That arc lies on the circle centered at Earth’s center (assuming that Earth is a perfect sphere). When two points have the same longitude, it means they lie on the same meridian, so the distance between them is measured along the meridian. Find on the Internet the approximate radius of the Earth and the coordinates of Toronto and Miami, and estimate the distance between these two cities.

11. Explain why the direct flight route from New York to New Delhi passes

close to the North Pole. 12. Suppose that for any points 1 1( , )A x y and 2 2( , )B x y on the plane we define

2 1 2 1( , )d A B x x y y= − + − . Show that ( , )d A B has all three properties of distance.

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13. Suppose that for any points 1 1( , )A x y and 2 2( , )B x y on the plane we define

{ }2 1 2 1( , ) max ,d A B x x y y= − − (the largest of the numbers, 2 1x x− and

2 1y y− ). Show that ( , )d A B has all three properties of distance. 14. Given two three-letter words, we can link them with a chain of words in

which only one letter changes from one word to the next. To go from BAT to MAN, for example, we can use the chain BAT => CAT => CAN => MAN. Let’s define the distance between two three-letter words as the length of the shortest chain that connects them (assuming that any two three-letter words can be connected by a chain). For example, the distance from BAT to MAN is equal to 2, because the chain BAT => MAT => MAN exists, and no shorter chain connects these words. Does this definition of distance on the set of all three-letter words turn this set into a metric space? In other words, is this a well-behaved definition of “distance”? Explain.

15. Let us define the distance between two polygons 1F and 2F on the plane as

the shortest distance between a vertex of 1F to a vertex of 2F . Is this a “well-behaved” definition of distance?

3.4 Relations and their Graphical Representations Mathematicians like to generalize. When they see examples of similar situations or phenomena they try to come up with a general abstract concept that captures the essence of the examples. They then define the concept formally and give it a name. They study the concept as a whole and apply it to other examples. The concept of a relation is one such very general abstract concept.

If A and B are two sets, a relation from A to B is a set of ordered pairs (a, b), where a is an element of A and b is an element of B.

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Example 1 Let L be a set of 3 letters: L = {‘a’, ‘b’, ‘c’}. Let W be a set of four words: W = {‘bat’, ‘cat’, ‘can’, ‘man’}. Let us consider a relation R, which is a set of all pairs (letter, word) such that letter occurs in word. R consists of seven pairs: R = {(‘a’,‘bat’), (‘a’,‘cat’), (‘a’,‘can’), (‘a’,‘man’), (‘b’,‘bat’), (‘c’,‘cat’), (‘c’,‘can’)}. This relation represents the relationship “letter occurs in word.” Note that more than one letter from L may occur in the same word, for example, (‘a’, ‘bat’), (‘b’, ‘bat’), and the same letter may occur in more than one word from W, for example, (‘c’, ‘cat’), (‘c’, ‘can’)).

Example 2 The set of all pairs of real numbers (x, y) such that 2 24x y= is a relation from to

.

The set of all pairs (a, b), where a is an element of A and b is an element of B, is called the Cartesian product of A and B and denoted as A B× . The word “product” is used, because when A and B are finite sets with m and n elements, respectively, then A B× has m n⋅ elements. In Example 1, L contains 3 letters, W contains 4 words, and L W× consists of all 12 possible letter-word pairs. By definition, a relation is a subset of A B× . The x-y coordinate plane can be thought of as × and any set of points on the plane is a relation from to . An interesting relation typically represents a particular relationship between elements of A and elements of B: a pair (a, b) belongs to the relation if the relationship of interest between a and b exists. That is why the term relation is used.

Often we consider relations from a set to itself. For example, A and B can be the same set of people.

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Example 3 Let S be the set of all Facebook subscribers, R the set of all ordered pairs 1 2( , )s s such that 2s is a “friend” of 1s . This relation is symmetric: if 1 2( , )s s is in R then

2 1( , )s s is also in R. In general, a relation from a set to itself does not have to be symmetric. The relation in Example 2 is not symmetric.

In this book we are primarily interested in relations from to . There are different ways to describe such a relation: in a table, as a plot, in words. But the most common and useful way is to use algebraic equations and/or inequalities.

Example 4 The set of all pairs (x, y) such that y x≥ is a relation.

Example 5 The set of all pairs (x, y) such that 2 and 1y x y≥ <= is a relation.

In an algebraic description of a relation, the word “and” is often replaced by a comma or by a curly brace. For example, 2 and 1y x y≥ ≤ can be written as 2 , 1y x y≥ ≤ or

2

1y xy

⎧ ≥⎨

≤⎩

This example is a system of inequalities. We can have a system of equations, too. Sometimes we mix equations and inequalities in one “system.”

As we know, in a Cartesian coordinate system there is a one-to-one correspondence between all points on the plane and their coordinates (x, y). So any relation on the set of real numbers can be represented as a set of points on the plane.

The graphical representation of a relation on the x-y plane is called its plot or graph.

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Example 6 Consider a relation between real numbers x and y defined by the inequality

2 2 4x y+ ≤ . Plot this relation on the coordinate plane. Solution In the geometric interpretation, this relation describes all points whose distance from the origin is less than or equal to 2. These are the points inside and on the border of the circle of radius 2 centered at the origin:

y

x 0

-2

2

2 -2

An equation typically describes a line or several lines. An inequality typically describes a region (or several disjoint regions). When a relation is described by a system of inequalities, its plot can be constructed as the intersection of the regions defined by the individual inequalities. It is sort of like a Venn diagram, only instead of random blobs we use precise regions.

Example 7 Plot 2 2 4, 0, 0x y x y+ ≤ ≥ ≥ .

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Solution

y

x 0

-2

2

2 -2

y

x 0

-2

2

2 -2 ⇒

Example 8 Plot the graph of the relation defined by a system of inequalities

2

2

y xx y

⎧ ≥⎪⎨

≥⎪⎩

Solution The graph of 2y x≥ consists of all the points on and above the parabola 2y x= . The graph of 2x y≥ consists of all the points on and to the right of the sideways parabola. The graph of the system of the two inequalities is the intersection of the two graphs. The two parabolas intersect at points (0, 0) and (1, 1), forming a “petal”:

y

x 0 1 -1

1

-1 With a little knowledge of calculus it is easy to show that the area of the “petal” is exactly one-third of the area of the unit square into which it is inscribed. But Archimedes obtained this result without calculus, in the third century BC. We will tell you how he did it in Chapter <...>.

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When a relation is described by equations connected with “or,” its graph can be constructed as the union of the regions defined by the individual relations.

Example 9 Plot the relation y x= . Solution y x= means y x= or y x= −

y

x

1

1

y = x y = -x

As we mentioned earlier, a relation does not have to be defined by formulas: it can be described in words, or by other means.

Example 10 Consider the set of all pairs (x, y), such that x and y are positive integers and y is evenly divisible by x. Plot this relation on the coordinate plane.

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Solution The plot consists of discrete points:

y

x

1

1 2 3 4 5 6

2

3

4

5

6

Cartesian coordinates revolutionized mathematics because they connected geometry to algebra: they provided a way to describe geometric figures with algebraic formulas and to visualize algebraic relations as geometric figures.

Example 11 A set of points on the x-y plane is described by the inequalities 2 5; 1 7x y≤ ≤ ≤ ≤ . What geometric figure is it? Solution It is a rectangle with the horizontal dimension 3 and the vertical dimension 6.

Exercises In Questions 1-16, plot the relation without using a calculator. 1. 2x y+ = 2. 1x y+ ≤ − 3. 1x y+ ≤ 4. 1x y− <

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5. 1x y+ ≤ 6. 3x ≤ 7. 3, 3x y≤ ≤ , where the points (x, y) are in the first quadrant 8. 3, 3x y≤ ≤ 9. 2y x> 10. 2 2,y x x y> > 11. 1xy ≥ 12. 1xy ≥ 13. ( 2)( 3) 1x y− + ≥ 14. 2 2 0x y+ = 15. 2 2 4 0x y y+ − = 16. ( )22 2 2 2x y y x y+ + = +

17.

y

x a

y

x

c

Give an example of a relation from to such that its graph is one vertical line. The same for one horizontal line.

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18. Come up with an inequality that describes all points on and inside the circle of radius 3 centered at (2, 0).

19. A relation on a set is called symmetric if with any pair 1 2( , )a a it also

contains the pair 2 1( , )a a . What is the geometric interpretation of a symmetric relation from to ? Which of the relations in Questions 1-16 are symmetric? How can we tell whether a relation described by equations and/or inequalities in x, y is symmetric, without plotting it?

20. Suppose a relation on real numbers with any pair (x, y) also contains the pair

(-x, y). Such a relation is not necessarily “symmetric” in the sense of the definition in Question 19, but its graph does possess a kind of symmetry. What kind? Which of the relations in Questions 1-16 have this kind of symmetry?

21. Suppose a relation on real numbers with any pair (x, y) also contains the pair

(-x, -y). What kind of symmetry does a graph of such a relation posses? Which of the relations in Questions 1-16 have this kind of symmetry?

22. Recall that in geometry the set of all points that satisfy a given condition is

called a locus of points (that satisfy the condition). Given two points, A and B, find the locus of points whose distance from A is twice the distance from

B (that is, all points P, such that 2PAPB

= ). Use coordinates but interpret the

result in geometric terms. Hint: Take the coordinate system with the x-axis directed along AB with the origin at A; find the two points 1P and 2P on the x-axis that belong to our locus of points; then shift the origin to the

middle between 1P and 2P , write an equation for 2PAPB

= in the new

coordinates and simplify it. 23. In Question 22, replace 2 with any positive number k. Draw the locus of

points P such that PA kPB

= for 15

k = , 12

k = , 1k = , 2k = , and 5k = .

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3.5 Polar Coordinates Cartesian coordinates are not the only way to establish a coordinate system on the plane. In many applications it is more convenient to use polar coordinates. For example, if a radar screen shows an airplane approaching the airport, the air traffic controller might be interested in the airplane’s distance from the airport and the direction from which the airplane is approaching. Polar coordinates use the distance of a point from the origin and the direction toward the point from the origin to describe the point’s location. In polar coordinates, distances are measured from one fixed point, called the pole. The pole is like the origin in the Cartesian system. The distance from a point to the pole is called the radius and is often denoted by r (Figure 3-3). Angles are measured from one base direction, called the polar axis. By convention, the polar axis is shown in graphs as a horizontal ray, starting at the pole and pointing to the right. It is like the right half of the x-axis. (Often the whole axis is drawn for convenience.) The polar axis also determines the units for measuring distances. Angles are measured in the counterclockwise direction, starting from the polar axis. The angular position of a point is often denoted by θ . The ( , )r θ pair serves as the polar coordinates of the point (Figure 3-3).

polar axis r

θ

1 2 3 4 0

pole

( , )P r θ

Figure 3-3. Polar coordinates

In practical fields, such as navigation and surveying, and in military applications, angles are measured in degrees. In mathematics and science, angles are often measured in radians.

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360 2π° = radians.

1 radian = 360 57.32π

≈ degrees.

Measuring angles in radians is often more convenient for mathematical formulas and scientific computations because the radian measure corresponds to the length of the arc of the unit circle (the circle of radius 1 centered at the origin) that spans the angle. So, if a particle is moving counterclockwise along the unit circle at a speed of 1 unit per second, the angle θ changes at the rate of 1 radian per second. On a Cartesian plane, points are plotted on a rectangular grid. For polar coordinates we use a polar coordinate grid, something like this:

3 0 2 1

2π

π

32π

0

3π

6π 5

6π

23π

116π

53π 4

3π

76π

Example 1

3 0 2 1

A

B C

D

What are the polar coordinates of points A, B, C, and D above?

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Solution

21,3

A π⎛ ⎞⎜ ⎟⎝ ⎠

, 73,6

B π⎛ ⎞⎜ ⎟⎝ ⎠

, 112.5,6

C π⎛ ⎞⎜ ⎟⎝ ⎠

, ( )2, 0D .

A ( , )r θ pair always defines a unique point on the plane. But the converse is not true: the same point can have multiple representations in polar coordinates. In fact, any point has an infinite number of representations in polar coordinates! First, any pair (0, )θ (with radius 0 and any value of θ ) represents the pole. Second, for any point, if we add or subtract 2π — the full revolution around the unit circle — to/from θ , we get the same point. In Example 1 above, we gave the polar coordinates of point D as ( )2, 0 , but ( )2, 2π , ( )2, 4π , ( )2, 100π , ( )2, 2π− , or

( )2, 100π− represent the same point. Third, the radius r can be negative! Convention permits negative radii in order to make the graphs of some polar functions consistent and continuous.

A negative radius r is interpreted as a positive radius with the same absolute value, but pointing in the opposite direction.

This means if we flip the sign of the radius and, at the same time, add π to θ (or subtract π from θ ), we get the same point. For example, ( )2,D π− represents the

same point as ( )2, 0D , and 13,6

B π⎛ ⎞−⎜ ⎟⎝ ⎠

represents the same point as 73,6

B π⎛ ⎞⎜ ⎟⎝ ⎠

.

Example 2

Plot on a polar grid 51.5,2

A π⎛ ⎞⎜ ⎟⎝ ⎠

, ( )2, 0B − , 11.5,3

C π⎛ ⎞−⎜ ⎟⎝ ⎠

, and ( )3, 6D π− .

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Solution

Equivalent simplified coordinates for these points are 1.5,2

A π⎛ ⎞⎜ ⎟⎝ ⎠

, ( )2,B π ,

41.5,3

C π⎛ ⎞⎜ ⎟⎝ ⎠

, and ( )3, 0D .

3 0 2 1

A

B

C

D

A polar coordinate system is related to a Cartesian coordinate system with the origin at the pole, the x-axis directed along the polar axis, and the same distance units:

r θ

1 2 3 4 0

( , )P r θ

-1 -2 -3 x

y

1

2

-1 It is not very hard to convert polar coordinates into related Cartesian coordinates. Let us first take a point ( , )P r θ in the first quadrant. Let M be its projection onto the x-axis.

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y

xO

P

M θ r

x

y

OPM is a right triangle with the hypotenuse OP . You might recall from geometry that in a right triangle the ratio of the length of the leg adjacent to θ to the length of the hypotenuse is called the cosine of θ and denoted cosθ ; the ratio of the length of leg opposite θ to the length of the hypotenuse is called the sine of θ ( sinθ ). In

other words (or rather in symbols), cosOMOP

θ= and sinPMOP

θ= . But OM is the

x-coordinate and PM is the y-coordinate of P in our Cartesian system. OP r= . We

get cosxr

θ= and sinyr

θ= . So

cos ; sinx r y rθ θ= =

Knowing θ , we can find cosθ and sinθ with a calculator.

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Example 3:

The polar coordinates of a point M are 5,7π⎛ ⎞

⎜ ⎟⎝ ⎠

. Find its Cartesian coordinates.

Solution

5cos7

x π= , 5sin

7y π= . After setting the calculator to radian mode, we find

cos 0.9009697π≈ ; sin 0.433884

7π≈ . So 5 0.900969 4.505x ≈ ⋅ ≈ ;

5 0.433884 2.170y ≈ ⋅ ≈ . The Cartesian coordinates of M are, approximately, (4.505, 2.170) .

What about points in other quadrants? The definitions of cosθ and sinθ are extended for all values of θ in precisely such a way that the above polar-to-Cartesian conversion formulas work. We will consider the general definitions of these “trig” (trigonometric) functions in Chapter <...>. Meanwhile, just remember that if you enter any value of θ in your calculator, get its cosine and sine, and plug them into the conversion formulas, the formulas will work. The formulas will work without any change even if r is negative!

Example 4:

Find the Cartesian coordinates of a point if its polar coordinates are 73,8π⎛ ⎞−⎜ ⎟

⎝ ⎠.

Solution

7 7( , ) ( 3)cos , ( 3)sin8 8

x y π π⎛ ⎞= − −⎜ ⎟⎝ ⎠

. Using a calculator we find

7cos 0.9238808π≈ − ; 7sin 0.382683

8π≈ . Therefore,

( )( , ) ( 3) ( 0.923880), ( 3) 0.382683 (2,772, 1.149)x y ≈ − ⋅ − − ⋅ ≈ − . The point lies in the fourth quadrant, as expected.

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Conversion from Cartesian coordinates into related polar coordinates is pretty easy,

too. We know that 2 2r x y= + . Also cos xr

θ = , and sin yr

θ = . We can use either

of these two equations to find θ , or we can use sintancos

yx

θθθ

= = . A graphing

calculator has commands for finding θ from a given value of a “trig” function by using the inverse function, such as 1cos− , 1sin− , or 1tan− . The only problem is that different values of θ can produce the same value of cos, sin, and tan, and the calculator gives you only one of them. If (x, y) resides in the first quadrant, the calculator always returns the correct value of θ ; otherwise you might need to adjust θ to match the quadrant in which (x, y) resides. This requires some understanding of how the “trig” functions behave, so let’s postpone the general case until Chapter <...>.

Example 5: Find the polar coordinates of M(4, 7). Solution

2 24 7 16 49 65r = + = + = . 7tan4

θ = . After setting the calculator to radian

mode, we find 1 7tan 1.0524

− ⎛ ⎞ ≈⎜ ⎟⎝ ⎠

(radians). Since M resides in the first quadrant, this

value does not need any further adjustment. The polar coordinates of M are

( )65, 1.052 .

A relation between r and θ (an equation, an inequality, or a combination of several equations and/or inequalities) can be plotted in polar coordinates, just like a relation between x and y can be plotted in Cartesian coordinates. Many simple polar equations produce interesting curves.

Example 6: Plot r θ= for 0θ ≥ .

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Solution

36

The curve r aθ= , where a is a positive constant, is known as the Spiral of Archimedes. In this example, 1a = . The spiral starts at the pole: r = 0 when 0θ = . The spiral crosses the polar axis at 2r π= , 4r π= ¸ 6r π= , 8r π= , and so on. An interesting question is how the spiral behaves around the pole. By definition, θ is the angle between the radius vector (the segment directed from O to P) and the polar axis. If you approach the pole along the spiral, θ becomes smaller and smaller, so the radius vector becomes more and more horizontal (even as it becomes shorter and shorter). At the pole, the spiral must be tangent to the polar axis. But you’ll need to zoom in really close to see that:

Google “polar curves” and you will find many web sites that show famous polar curves. Many sites have animations and Java applets, which let you change the parameters of the equation or let you see plotting in progress. Follow the link, for example, to “Famous Curves Applet Index.” Many of these curves are defined by equations that use trigonometric functions, so we will take another look at them later, in Chapter <...>. Questions <...>-<...> in the exercises ask you to plot a couple of simple polar regions and lines.

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Exercises 1. Look up Archimedes’ life story on the Internet. Why was his tomb decorated

with a sphere inscribed into a cylinder? 2.

3 0 2 1

A

B

C

D

Determine the polar coordinates of the points A, B, C, and D above (with r > 0 and 0 2θ π≤ < ).

3. On a polar coordinate grid, plot the points with the following polar

coordinates: (0, 0) , 2,2π⎛ ⎞

⎜ ⎟⎝ ⎠

, 51,6π⎛ ⎞

⎜ ⎟⎝ ⎠

.

4. On a polar coordinate grid, plot the points with the following polar

coordinates: 71,2π⎛ ⎞

⎜ ⎟⎝ ⎠

, 3,2π⎛ ⎞−⎜ ⎟

⎝ ⎠, 3,

3π⎛ ⎞−⎜ ⎟

⎝ ⎠.

5. Plot the following points in polar coordinates, then convert them into

Cartesian coordinates.

(a) ( )1, 0 ; (b) 1,2π⎛ ⎞

⎜ ⎟⎝ ⎠

; (c) ( )1, π ; (d) 31,2π⎛ ⎞

⎜ ⎟⎝ ⎠

.

6. Convert the following polar coordinates into Cartesian coordinates without

plotting the points.

(a) 2,6π⎛ ⎞

⎜ ⎟⎝ ⎠

; (b) 51.5,8π⎛ ⎞

⎜ ⎟⎝ ⎠

; (c) ( )5, 6.2π ; (d) 1,7π⎛ ⎞−⎜ ⎟

⎝ ⎠.

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7. Convert the following Cartesian coordinates of points in the first quadrant

into polar coordinates (with r > 0 and 02πθ≤ ≤ ):

(a) ( )1, 3 ; (b) 23,7π⎛ ⎞

⎜ ⎟⎝ ⎠

; (c) ( )2, 2 ; (d) ( )1, 7 .

8. What is greater: θ (in radians) or sinθ , for 02πθ≤ ≤ ? Hint: consider the

Cartesian coordinates of (1, )θ . 9. Plot on a polar grid the region defined by the system of inequalities

2 3,6 3

r π πθ≤ ≤ ≤ ≤ .

10. Write an equation in polar coordinates that describes the circle of radius 1

centered at the pole. 11. Write an equation in polar coordinates that describes the vertical line through

the point ( )3, 0 . Hint: write the equation in Cartesian coordinates first, then convert it to the polar form.

12. Write an equation in polar coordinates that describes the vertical line through

the point ( )5, π . 13. Write an equation in polar coordinates that describes the horizontal line

through the point 4,2π⎛ ⎞

⎜ ⎟⎝ ⎠

.

14. Rewrite the equation 2 2 4 0x y y+ − = (from Question <...>(15) in

Section <...>) in polar coordinates. 15. Plot the Spiral of Archimedes for 0θ ≤ .

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16. Using geometry, plot on the polar coordinate plane the relation cosr a θ= , where a is a positive constant.

Hint:

a

17. Plot the relation from Question 16, using algebra. Hint: multiply both

sides by r, then convert to Cartesian coordinates, then shift the origin to

, 02a⎛ ⎞

⎜ ⎟⎝ ⎠

and rewrite the equation in new coordinates.

18. We explained in Example 6 that the Spiral of Archimedes is tangent to the

polar axis at the pole. The curve cosr a θ= also goes through the pole (see Questions 16 and 17), but it is not tangent to the polar axis. In fact, it has a vertical tangent line. How can this be? Explain the difference between these situations.

3.6 Review Concepts, terms, methods, and formulas introduced in this chapter:

Cartesian coordinates x- and y-axes Origin Quadrants Coordinates in the new coordinate system when the origin shifts to (a, b):

x x ay y b= −= −

Distance between 1 1( , )A x y and 2 2( , )B x y : 2 22 1 2 1( ) ( )d x x y y= − + −

Distance properties The triangle rule Metric space Relation System of equations or inequalities Graph of a relation Polar coordinates Pole Polar axis

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Radians; 360 2π° = (radians) ( , ) ( , 2 )r rθ θ π= ± ( , ) ( , )r rθ θ π− = ±

θ r

x

y cos x

rθ =

sin yr

θ =

2 2

tan

r x yyx

θ

= +

=

From polar to Cartesian coordinates: cosx r θ= , siny r θ= .

From Cartesian to polar coordinates: 2 2r x y= + , 1tan yx

θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠

.

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3.1 prologue1 3 coordinates 3.1 prologue coordinates describe the location of a point or an object...

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