Vectors and Parametric Equations

V ^ECTORS

AND

M ^OTION

Motion is a pervasive aspect of our lives. You walk and travel by bike, car, bus, subway, or perhaps even by boat from one location to another.

You watch the paths of balls thrown or hit in the air and of space shuttles launched into orbit. Each of these motions involves both direction and distance. Vectors provide a powerful way for representing and analyzing motion mathematically.

In this unit, you will learn how to use vectors and vector operations to solve problems about navigation and force. You will extend and further connect your understanding of

geometry, trigonometry, and algebra to establish properties of vector operations and to create and use parametric equations to model linear and nonlinear motion.

The key ideas will be developed through work on problems in three lessons.

Lessons

UNIT

2

UNIIT IT

Linear Motion

Develop skill in using vectors, equality of vectors, scalar multiplication, vector sums, and component analysis to model and analyze situations involving magnitude and direction.

Vectors and Parametric

Equations Simulating Nonlinear

Motion

Use parametric equations to model nonlinear motion, including the motion of projectiles and circular and elliptical orbits.

Represent and analyze vectors and vector operations using coordinates.

Use position vectors to develop parametric equations to model linear motion.

LESSON 1 Linear Motion

Each day you confront motion in nearly everything you do. You may walk, ride a bicycle, or ride in a car or bus to school. You may take a subway train to meet friends at a shopping mall. You see aircraft fly overhead and you see the position of the Sun in the sky move, from morning when it rises in the east to evening when it sets in the west. You might run in a race or throw, kick, or hit a ball.

In this unit, you will learn to use an important tool for modeling motion—vectors. Vectors are useful in situations that involve magnitude (such as distance) and direction. These are important descriptors of motion. The simplest motion is linear—movement along a straight line.

Linear motion is used to plan as well as guide hiking routes and courses of boats and ships. Think about how you might describe or represent a planned route on a map. Think also about conditions that might affect a planned route and how you might incorporate that information in the planning process.

Think About This Situation

Suppose you wanted to map out a route that involved sailing 3 km west from Bayview Harbor to Presque Island, then 6 km south to Rudy Point, and then 5 km southeast to Pleasant Bay.

How could you represent the planned route geometrically?

How could you represent a direct sailing route from Bayview Harbor to Pleasant Bay?

How could you estimate the length of the route in Part b? How would you describe its direction?

How would a northeast water current affect the path along which you would steer the boat to maintain the route in Part b?

In this lesson, you will learn how to represent vectors geometrically, how to scale vectors, and how to combine vectors by addition. You will also explore how vectors can be used to solve applied problems.

Navigation: What Direction and How Far?

Vectors and vector operations are used extensively in navigation on water and in the air. As you work on the problems of this investigation, look for answers to these questions:

How can vectors be represented geometrically with directed line segments?

How can vectors and scalar multiples of vectors be used to model navigation routes?

Charting a Boat’s Course Imagine that you are navigating a boat along the small portion of the Massachusetts coast shown in the nautical chart at the right.

Note that within the chart itself, there are several aids to

navigation such as buoys, landmarks, and scales. The buoys are painted red or green and may have a red or green flashing light. A circle (on land) with a dot at its center indicates an easily recognized landmark

like a stone tower or a tank. Adapted from Frank J. Larkin. Basic Coastal

As a class, examine a copy of the nautical chart.

a. At the right of the chart is a nautical mile (nm) scale. Use this scale to find the distance from
the “SH” buoy to the “GP” buoy. Measure between the centers of the circles that mark the buoys.

b. There are other scales at the top and along the right edge of the chart. What do you think these scales represent?

c. What other scale on this chart can be used to measure nautical miles? What does a nautical mile represent based on this scale?

d. A nautical mile is 6,076.1033 feet. How does a nautical mile compare to a statute mile (regular mile)?

Coastal water nautical charts are designed so that the top is due north and the right side is due east. You can use your knowledge of directed angles measured counterclockwise from the horizontal (due east) to describe the direction of a craft. Thus, you can say due east is 0˚, due north is 90˚, due west is 180˚, and due south is 270˚.

The course of a boat starting at Buoy
6 and moving 30˚ north of east is shown in the chart at the right. Use your copy
of the nautical chart to complete this problem. Using a ruler made from the nautical mile scale, measure distances to the nearest ₁₀¹
nm. Measure angles to the
nearest degree using a protractor.

a. Mark and label a point P on a copy of the chart to represent a boat that is 3
nautical miles from the

“3” bell and is headed at an angle of 290˚. What buoy is nearest to P?

b. Draw an arrow from the “SH” buoy to the “6” buoy. What is the direction in degrees? What is the distance in nautical miles?

c. What are the direction and distance of the path from the “6” buoy to the center of the mouth of the channel at Stone Harbor?

d. Why are arrows particularly useful representations for nautical paths?

The arrows that indicate boating routes are directed line segments. They have both a magnitude (length) and a direction. Thus, an arrow is a geometric representation of a vector—a quantity with magnitude and direction. A vector with a length of 1" and direction of 45˚ is shown at the right.

a. Accurately draw arrows representing vectors with the following characteristics.

i. Length: 2 nm; direction: 70˚ (Use your nautical ruler and a protractor.) ii. Length: 5 cm; direction: 110˚

iii. Magnitude: 7 cm; direction: 300˚

b. Draw an arrow for each vector described. State what length you chose to represent 1
knot and what length you chose to represent 1
mph.

i. A boat with speed of 2 knots (nautical miles per hour) at a direction of 180˚

ii. Speed of 40 mph at a direction of 240˚

iii. Force of a 15 mph wind blowing from the northeast

c. Compare the arrows you drew in Parts
a and b with your classmates. Resolve any differences.

Denoting Vectors Vectors can be denoted in various ways. One way is to use italicized letters with arrows over them, such as

_{a or}

_v . When the initial point, or tail, A and terminal point, or head, B are labeled, notation AB can be used. Since a vector

_v is determined by its magnitude r, and its direction
,

_v can also be represented as a pair, [r,
]. (The square brackets in this case do not represent a closed interval of numbers.)

Since arrows representing vectors, as in Problem 3, can be drawn anywhere in a plane, it is important to know that two arrows drawn using the same scale represent equal vectors when they have the same magnitude and the same direction.

Explain why the following method provides a geometric test for the equality of the vectors _{PQ and}

_{RS .}

Step 1. Connect the heads Q and S and connect the tails P and R.

Step 2. If PQSR is a parallelogram, then _PQ
=

_{RS .}

Scalar Multiples of a Vector In the problems that follow, use either a ruler, protractor and graph paper, or interactive geometry software with vector drawing and analysis capabilities. It may also be helpful to use the “Stone Harbor” custom tool in CPMP-Tools. When the instructions ask you to make an “accurate drawing” or an “accurate sketch” of a vector, you can do so with geometry software or carefully draw an arrow on graph paper using a ruler and protractor to measure. If, however, the instructions are simply “sketch” or “draw” a vector, you may make a freehand sketch of an arrow that approximates the characteristics of an accurately drawn vector in order to guide your thinking. Note that “draw a vector” actually means “draw a geometric representation of the vector” (an arrow).

A fishing boat leaves the mouth of the Stone Harbor channel trolling on a heading due north at a speed of 1.5
knots (nautical miles per hour).

a. On your copy of the nautical chart, sketch the vector

_v representing the distance and direction traveled from the middle of the channel opening during the first hour.

b. Use the vector

v in Part a to determine the vector for a 2-hour trip at the same speed and in the same direction. Sketch this vector. Label it 2

_{v .}

c. Sketch and label a vector that locates the fishing boat at the end of 20 minutes. At the end of 2.5
hours.

d. Now sketch another vector that has the same length as 2

v but is not equal to 2

_{v , and}

another vector that is equal to 2

v . Compare your vectors with those of your classmates.

e. In general, how would you sketch a vector that was a positive number k times a given

Suppose another boat begins a trip at the same point at the mouth of the channel at Stone Harbor headed at a direction of 20˚ and at a speed of 2 knots.

a. Sketch the vector

_v showing the approximate position at the end of the first hour.

b. Suppose the boat returns to the harbor along the same route at the same speed.

Sketch the return vector and give its magnitude and direction.

c. The word “opposites” can be used to denote the vectors in Parts a and b. How is the word “opposite” descriptive of the relationship between the two vectors?

d. Sketch a vector opposite to vector

_v in Part a with initial point at the “3” bell. Give its magnitude and direction.

When a vector

a is multiplied by a real number k, the number is called a scalar and the product, k

a , is a scalar multiple of

a . (In a similar manner, kAB is a scalar

multiple of the vector AB .) When k > 0, k

a is the vector whose length is k times the length of

a and has the same direction as

a as shown at the right.

For vector

a shown above, the opposite of vector

_{a ,}

denoted –

a , is shown at the right. The scalar multiple k

_a

when k < 0 is shown at the far right.

a. Compare the relationship between

_{a and k}

a when k>0 to the relationship between –

a and k

a when k<0.

b. Suppose

a = [10, 50˚] and

_v = [8, 20˚]. Write each of the following vectors in the form [r,
] where r is the vector’s magnitude and its direction
satisfies 0˚

< 360˚.

i. ⁵

_{a ii.}

^0.2_v

iii. –

_{a iv.}

^–3

_v

c. Suppose k < 0 and

_{a = [r,}
]. Write k

a as a [magnitude, direction] pair in terms of r and
.

Summarize the Mathematics

In this investigation, you explored how vectors—quantities with magnitude and direction—

can be represented geometrically by arrows.

Describe how you know when two arrows represent the same vector.

How are a vector and a scalar multiple of that vector similar? How are they different?

How are vectors _AB

_{and BA}

alike? How are they different? What is another way to write _BA

using A and B?

What is always true about the magnitudes and directions of two opposite vectors?

Be prepared to explain your ideas to the class.

Daily ferries shuttle people and cars between Manitowoc, Wisconsin, and Ludington, Michigan.

Use
a copy of this map of Lake Michigan to complete the following tasks.

a. Draw the vector for the ferry route from Manitowoc to Ludington. Label it

_{v .}

Measure its magnitude and direction.

b. Find the magnitude and direction of –

_{v .}

Draw – v beginning at Charlevoix,

Michigan.

c. Sketch ^0.5_v

from Milwaukee, Wisconsin.

Find its magnitude and direction.

d. Are the vectors representing the route from Charlevoix to Escanaba and the route from South Haven to Milwaukee approximately the same? Explain.

Changing Course

In the previous investigation, you used vectors to model straight-line paths. As you complete the problems in this investigation, look for answers to the following question:

How can vectors be used to model routes when there is a change of course during the trip?

Vector Sums Roberta, the skipper of the fishing boat High Hopes, leaves the mouth of the Stone Harbor channel making 6 knots at 25˚. She travels for 20 minutes, then turns so that she is heading in a direction of 100˚ at the same speed and travels for 30 minutes before deciding to drop anchor and begin fishing.

a. Using the “Stone Harbor” custom tool or a copy of the nautical chart, draw a vector diagram showing the paths taken and the position of the High Hopes at the end of 50
minutes. What is the

magnitude of each of these two vectors?

b. Suppose the fish are biting and Roberta wants to inform Clarissa, the skipper of the Salmon King, of where she is located so Roberta can join her. Draw a vector representing the path Clarissa should take from the mouth of Stone Harbor channel directly to the High Hopes. What direction should Roberta advise her to take? How far will she need to travel?

c. The vector representing the path that Clarissa should travel to the good fishing spot is called the sum or resultant of the two vectors that describe the route taken by the High Hopes.

How are the initial and terminal points of the resultant vector in Part b related to the two vectors that represent the route taken by the High Hopes?

d. Suppose Roberta had left the harbor at a speed of 6 knots in a direction of 100˚ for 30
minutes
and then turned to a direction of 25˚ and traveled for 20 minutes at the same speed. Draw a sketch of Roberta’s path and the resultant vector. What are the direction and the magnitude of the resultant vector?

e. Now sketch two additional two-leg routes to this good fishing spot.

f. For each of the following statements, decide if it is true or false. In each case, explain your reasoning.

i. The vector sum of any two given vectors is unique.

ii. If a vector is the sum of two given vectors, it cannot be the sum of two other vectors.

For the following vectors, the magnitude is in centimeters and the given angle measure is the vector’s direction:

a = [5, 70˚],

b = [4, 30˚],

_c = [4, 350˚], and

d = [3, 250˚]. Make accurate drawings of each vector sum and measure to find the magnitude (to the nearest 0.1 cm) and direction (to the nearest 5˚) for each resultant vector.

a.

_a+

b b.

_a+

d c.

_a+

b +

_c

Now investigate some general properties of vector addition. Begin by sketching any two vectors

a and

b as arrows that have different directions but no points in common.

a. Draw a diagram showing how to place

_{a and}

b to find

_a+

b. Do the same for

b +

_a.

What do you notice about the two vector sums? Compare your observations to those of others and resolve any differences.

b. To which property of real number operations is this similar?

c. Choose a point in the plane. Place

_{a and}

b so their initial points are at this point. Then draw a single diagram showing how to find

_a+

b and

b +

_a. What shape is formed? Prove your conjecture.

On a sheet of plain or graph paper, make an accurate drawing of a vector _u

with magnitude 4
cm and direction 200˚ and a vector

_v with magnitude 5 cm and direction 70˚.

a. Without drawing or measuring, find the magnitude and direction of as many of the following vectors as possible. Explain your reasoning in each case.

i. ²_u

_ii.

_{v +u}

iii. _u

₊

_{v iv.}

³⁽_u

₊

_{v )}

v. ³

_{u+ 3v}

_vi.

–2

_v

vii. ²

_{v + (
2u )}

_viii.

²_v

_{+ (
2}

_{u )}

b. Find the magnitude and direction of the remaining vectors by measuring on paper. Use as few drawings
as possible. Look for possible connections between pairs of vectors that might reduce your work.

c. What general rule is suggested by parts iv and v in Part
a? Test your conjecture.

Horizontal and Vertical Components The chart below shows a vector

_v with magnitude 1.3 nm and direction
0˚ and a vector _w

with magnitude 1.8
nm and direction 90˚ that give one route to a good fishing area.

a. Calculate (do not measure) the magnitude of the resultant vector _v

_+w

_.

b. Use trigonometric ratios to compute the direction of the direct route

_{v +w}

_{to the}

good fishing
spot.

c. Starting at the harbor, is it possible to find another pair of vectors with directions 0˚

and 90˚ that have the same vector sum as in Part
a?

Explain your reasoning.

Now investigate further how a vector can be thought of in terms of the sum of horizontal and vertical vectors called its components.

a. Suppose a vector represents a 2-nautical mile route with a direction of 78˚. Use

trigonometric ratios to compute the lengths of the east (0˚) and north (90˚) legs of a route to the same location.

b. Suppose a vector

_v represents a 2-nm route with a direction of 125˚. Make a sketch of the vector

v and include the west and north vectors that would give the resultant vector

_{v .}

Compute the magnitudes of the west and north vectors.

c. Now think more generally. How would you compute the magnitudes of the horizontal and vertical components of the vectors described below? Compare your methods with those of your classmates and resolve any differences.

i. Any 2-nm vector with a direction
between 180˚ and 270˚

ii. Any 5-nm vector with a direction
between 270˚ and 360˚

iii. Any 10-nm vector that points due north or due south iv. Any 10-nm vector that points due east or due west

Summarize the Mathematics

In this investigation, you explored the geometry of addition of vectors.

Describe geometrically how you can find the resultant, or sum, of two vectors.

Any nonzero vector can be represented as the sum of a horizontal vector and a vertical vector.

Illustrate and explain how this can be done for a given vector.

In the vector diagram below, _AC

_and

_CB

are the horizontal and vertical components, respectively, of _AB

_.

i. If you know the magnitudes of _AC

_and

_CB

_,

how would you calculate the magnitude and direction of _AB

_?

ii. If you know the magnitude and direction of _AB

, how would you calculate the magnitudes of _AC

_and

_CB

_?

Be prepared to share your ideas and reasoning with the class.

Use what you have learned about adding vectors and horizontal and vertical components of a vector to compute (not measure) answers to the questions below. Check that your answers are reasonable by measuring.

a. Suppose Clarissa wants to fish in the secluded bay behind Great Point, as shown in the chart.

The vector [3.1 nm, 10˚] represents a direct route to the bay. Since this route crosses land, Clarissa decides to head east and then due north to the fishing spot. How many nautical miles should she travel east before turning north? How far north from there is the fishing spot?

b. Suppose Roberta needs to travel from her location south of Hog Island to the west side of Oak Island before nightfall. The west and north vectors for one route are shown on the chart. If Roberta decides to take a direct route (across the rocky area) rather than the west/north route, how many nautical miles can she shave off the trip?

In what direction should she head?

Go with the Flow

The vector models you have been using for navigation assume that the force moving a boat is the only one acting on the craft. When this
is the case, the craft moves in a straight line in the direction of
the force. In reality though, two (or more) forces often act simultaneously on an object.

For example, currents in the ocean are forces on boats that move the boats in the direction of the current. Sailing ships without motors use water currents to help them enter and leave port. The wind, too, is a force that affects the path that a boat or an airplane follows. A fundamental principle of physics is that the effect of two forces acting on a body is the sum of the forces. As you work on problems in this investigation, look for answers to this question:

How can vectors be used to analyze the effect of two or more forces acting

on an object simultaneously?

Suppose a boat leaves port P headed in a direction of 60˚ with the automatic pilot set for 10 knots. On this particular day, there is a 4-knot ocean current with a direction of 30˚. The vector diagram at the right shows the effect of the current on the position of the boat at the end of one hour.

a. Assuming a scale of 1 cm = 2 nm, verify the accuracy of the diagram.

b. The sum of the course and current vectors gives the position of the boat in one hour.

Determine how far the boat will actually travel in one hour:

i. using the scale diagram.

ii. using the Law of Cosines. (Hint: The obtuse angle of the triangle is 150˚. Why?) c. At what speed and in what direction will the boat actually travel during the first hour?

Would it continue to travel similarly during the next hour if all conditions remained the

In Problem 1, you were able to determine the actual course of the boat using either a scale drawing and measurement or using the Law of Cosines. Now examine the situation in terms of component vectors.

a. Make a sketch similar to the one in Problem
1 that shows the planned-course vector and the current vector without the resultant vector. Then sketch the horizontal and vertical

components of each vector.

b. Compute the lengths of the horizontal and vertical components of each vector in your sketch from Part
a.

c. Using the component vectors found in Parts a and b, find the components, magnitude, and direction of the resultant vector representing the actual route.

d. Compare the results of this problem with those of Problem 1.

Consider vector

_v , with length 4 cm and direction 35˚, and vector _u

with length 5 cm and direction 70˚.

a. On graph paper, draw

_{v +u}

_.

b. Draw the horizontal and vertical components of

_{v and of}

_u.

c. Draw the resultant of the horizontal components and the resultant of the vertical components.

d. Draw the sum of the two resultant vectors found in Part c. How is this sum related to

_{v +}

_u?

e. Describe how the components of two vectors can be used to find the sum of the two vectors.

Make a sketch of a vector showing the location of an airplane traveling at a fixed altitude after one hour if it is headed in a direction of 40˚ and its speed in still air is 500 mph, but the wind is blowing at 40
mph from the northwest.

a. Augment your sketch to show the horizontal and vertical components

h_v and

_v

v of the planned-course velocity vector _v

. On the sketch, represent the horizontal and vertical components

h_w and

_v

w of the wind velocity vector _w

_.

b. Use the component vectors in Part a to determine the direction and distance the airplane traveled in one hour.

c. What was the effective speed of the airplane during that one hour of flying? (This is called the ground speed.)

Combining Forces The process illustrated in Problems 2–4, called component analysis of vectors, is a very powerful tool for analyzing linear motion problems. It reduces a complex situation to one in which only component vectors with the same direction are added. The next two problems will provide you with further practice in using component analysis to solve applied problems.

Two men have to move a doghouse on skids to a new position due east of its present location.

They tie ropes to the doghouse and pull as follows: Thad pulls with a force of 100 pounds at a direction of 35˚, while Jerame pulls with a force of 120 pounds at a direction of 315˚.

a. Make a sketch showing the force vectors and resultant vector. Ignore the force of friction.

b. Find the direction at which the doghouse should move under these conditions.

c. If friction requires 150 lb of force to move the doghouse, will it move with the given effort of Thad and Jerame? Explain.

d. Jerame and Thad wanted the doghouse to slide due east. Thad suggested that Jerame change the direction at which he pulls so that Jerame’s south force cancels out Thad’s north force.

Would the suggestion work? Why or why not?

e. If Thad pulls as before, in what direction should Jerame pull to slide the doghouse due east when they both pull it?

In Problem 1, you found that ocean current causes a boat to travel off the desired course. As shown in the first diagram at the right, the resultant vector

_v is the sum of the desired course of the boat

b and the current vector

_{c .}

Suppose the captain of the boat wants to stay
on the desired course

b
=
[10
knots,
60˚]

by
adjusting for the current vector

_c
=
[4
knots,
30˚]. In the second diagram at the
right, vector

_x represents this adjusted course
setting.

a. How can

_x be expressed in terms of

b and

_{c ?}

b. Explain why this adjusted course path will give the desired course [10
knots,
60˚].

c. What are the magnitude and direction of

_x?

Summarize the Mathematics

In this investigation, you examined how vectors can be used to analyze situations in which more than one force is acting on an object.

Describe how vectors can be used to model linear motion in moving air or water.

Explain how the horizontal and vertical components of vectors can be used to determine the direction and speed of a boat or airplane that is moving at a fixed speed along a linear path in water or air that is also moving at a fixed speed along a linear path.

Be prepared to share your descriptions and thinking with the class.

The pilot of a commercial jet airplane wants to fly in the direction of 70˚ and average 600 mph, but a 70 mph wind is blowing from the northwest.

a. Draw a vector model of the effect of the wind on the jet.

b. Draw a vector model showing the direction needed to keep the jet on course. Compute the direction.

c. Compute the still air speed that the jet needs to maintain to attain the desired average of 600
mph.

Tony Hillerman was a mystery writer whose books are often based on the native American cultures of New Mexico, Utah, Colorado, and Arizona. The map below shows Hillerman country in which Navajo Tribal Police Officers Joe Leaphorn and Jim Chee solve mysteries. In

Hillerman’s novels, they travel mostly by car throughout the reservations, but for this task assume they have a helicopter. Use careful sketches to measure the desired magnitudes and angles.

Adapted from “Recreation Map of Arizona and the Four Corners Region,” North Star Mapping.

a. Suppose Jim and Joe are stationed at Shiprock. What direction should Jim chart to go to Tuba City to investigate a hit-and-run accident? What is the distance he must fly by helicopter?

b. Jim is to fly from Tuba City to Flagstaff to meet with FBI officials. In what direction is he headed? At 100 mph, what is his flying time?

c. Plot the round trip from Shiprock to Round Rock to Window Rock to Standing Rock and back to Shiprock. Give the direction and distance of each part of the trip.

Suppose a boat leaves harbor at noon in still water at a direction of 310˚ and a speed of 4.4
knots, and vector

_v represents the distance and direction traveled during the first hour of the
trip.

a. Label a point on your paper as the harbor. Use a scale of 1 cm = 1 knot. Sketch vector

_{v ,}

and write it as a [magnitude,
direction] pair.

b. Sketch the vector that represents the position of the boat at 1:30 P.M. Write this vector in terms of

_v and as a [magnitude, direction] pair.

c. At 3:00 P.M., the skipper turns the boat back toward the harbor, maintaining a speed of 4.4
knots. Make a sketch of vector _w

that represents the distance and direction that the boat travels from 3:00
P.M. until 5:00 P.M. Then write this vector as a [magnitude,
direction] pair.

A Coast Guard cutter is located in a harbor in

Ludington when an SOS comes in from a boat located at a point due west of Grand Haven and due south of Manitowoc. Grand Haven is about 62
miles from Ludington at a direction of 280˚, and Manitowoc is about 60
miles from Ludington at a direction of 170˚.

a. Find the components of the vector from Ludington to Grand Haven.

b. Find the components of the vector from Ludington to Manitowoc.

c. Use a copy of the map at the right as a guide to sketch the components that you found in Parts
a and b and the location of the boat that is in trouble.

d. Using your sketch as a guide, calculate the magnitude and direction of the vector from Ludington to the location of the boat.

e. Suppose a second Coast Guard cutter is located at Manitowoc. Assuming this cutter can travel at the same speed as the first, which boat should respond to the SOS?

Any vector can be thought of in terms of its horizontal and vertical components. Similarly, if you know the components of a vector, you can determine the vector’s magnitude and direction.

a. Calculate the directed lengths of the horizontal and vertical components of the following vectors written in [magnitude, direction] form.

i. [3 nm, 86˚] ii. [5 nm, 285˚]

iii. [2.5 nm, 120˚] iv. [6 nm, 315˚]

b. Calculate the magnitude and direction of the vectors with the following components.

i. 5 nm east, 3 nm north ii. 12 nm east, 7 nm south iii. 7 nm west, 1 nm north iv. 21 nm west, 6 nm south

The nautical chart used in Investigation
1 is reproduced at the right. The boat Open C is located at the flashing red light at Sunken Ledge when its skipper learns that fishing action has begun near the “GP” buoy.

a. In what direction should the skipper head for the “GP” buoy?

b. At 6 knots, how long would the trip take in still water?

c. Now suppose there is a heavy northeast wind that will move boats at a rate of about 2
knots.

Make a vector diagram showing the effect of the wind on the course of the Open C.

d. In the wind, what is the direction of the route the Open C actually travels?

e. What direction should the skipper plot to account for the wind and follow a direct route to the “GP” buoy?

A balloon ride can be a very beautiful and peaceful experience, but balloon operators must always be on guard for the effects of the wind. Suppose a balloon rises at a constant rate of 2.4
meters per second, but there is a wind blowing at 1 meter per second from the west.

a. Taking the effect of the wind into account, what are the speed, direction, and components of the balloon’s velocity vector?

b. How high above the ground will the balloon be after 5 minutes?

c. After 5 minutes, the balloon is directly above a monument on the ground. How far is the monument from the point at which the balloon originally ascended?

Maria and Kim are volleyball players on their

respective school teams. Suppose that in a conference match, at the same time, they each block the ball when it is directly over the net. Maria’s hit has a force of 50
pounds at a direction of 325˚. Kim’s hit has a force of 40 pounds at a direction of 60˚.

a. Sketch the vectors involved if the net is on the east-west line.

b. Assuming that the ball moves in the direction of the resultant force, on whose side of the net will the ball land? How can component vectors be used to verify this?

c. At what angle should Maria hit the ball so that it follows the top of the net or goes onto Kim’s
side?

In each of the diagrams below, a figure F and its image G under a translation are shown.

Translation I Translation II

a. How could you use a vector to describe each translation?

b. Can every translation be described by a vector? Explain your reasoning.

c. Sketch any triangle
RST and its image under the composition of two translations, the first with vector _u

= [3 in., 90˚] and the second with vector _w

= [3 in., 180˚]. In your drawing, also sketch the single vector that represents the composition of the two translations.

d. Describe how the vector representing the composite transformation in Part
c is related to the two translation vectors, _u

_and

_w

_.

In Investigation
2, you added two vectors _{u and}

v by placing vectors head-to-tail as shown below. If AB represents

_{u and}

BC represents

_{v , then u}

₊

_{v = AC .}

a. The parallelogram law provides a second way to add the two vectors _{u and}

_{v .}

Explain as precisely as you can why by completing the parallelogram ACDB, the diagonal vector AD equals

_u+

_{v .}

b. Suppose

a = [3 cm, 20˚] and

b = [5 cm, 50˚]. Use the parallelogram law to find
^{a +
b .} Express the sum in [magnitude, direction] form.

Vector addition and scalar multiplication have some of the same properties as real number addition and multiplication.

a. Using the vectors shown at the right, make sketches illustrating the following properties.

i.

_a+

b =

b +

_a

ii. ⁽

_{a +}

b ) +

_{c =}

_{a+ (}

b +

_{c )}

iii. ²⁽

b +

_{c )= 2}

b + 2

_c

b. For any two real numbers, s and t, s
–
t
=
s
+
(–t). The expression _{a –}

b , by definition, is the vector that when added to

b gives the resultant

a , that is, ⁽

_{a –}

b )+

b =_{a . The}

expression

_{a + (–}

b ) represents the sum of vectors

_{a and –}

b . Using the vectors above, sketch

_{a –}

b and

_{a + (–}

b ) . Show that the processes you use to make the sketches are different, but the vectors that result are equal vectors.

Consider the partial tiling of equilateral triangles shown at the right. Write expressions for each of the following vectors in terms of

_{u and}

_{v .}

a. _AB

_{b. BC}

c. _AC

_{d. BD}

Using the diagram below and properties in Task
10, write each expression in a simpler form.

a.

b –

_u

b. _{w –}

_a

c.

_{v +u –}

b d.

b +

_{a –}

_{v –w}

On a piece of paper, mark a point O in the center. Using O as the beginning point, accurately draw a 1-inch vector,

a , pointing horizontally to the right and accurately draw a 1.5-inch vector,

b , pointing straight upward.

a. What is the measure of the angle between these vectors?

b. Choose a point P on your paper so that the length of OP is 4 inches. Find scalars m and n so

that _OP

_{= m}

_{a+ n}

b .

c. If Q is any other point, can you always find scalars m and n such that m

_{a+ n}

b = _{OQ ?}

Explain your reasoning.

d. Revise your answer for Part c when OQ has the same direction as

_{a or}

b . When Q coincides with point O.

Wind patterns over a region are sometimes plotted using vectors that show the direction and force of the winds at various locations. The wind patterns over the San Francisco Bay on August 17, 2006 are shown on the chart at the right.

a. Describe where the wind is the strongest. What is the direction of the strongest winds?

b. Describe locations where the wind is blowing from the west. From the south. From the north.

c. Describe two separated locations in which the wind vectors are approximately equal.

Approximately opposite. What does this mean about the wind in those locations?

For two vectors to be equal, two conditions must be met: their lengths must be equal and their directions must be the same.

a. Sketch vectors to illustrate the necessity of both conditions by:

i. showing that two vectors with the same length may not be equal.

ii. showing that two vectors with the same direction may not be equal.

b. If two vectors are equal and begin at the same point, how are their geometric representations
(arrows) related?

c. If two vectors are equal and begin at different points, how are their geometric representations
related?

The following questions will help you refine your thinking about the sum, or resultant, of two
vectors.

a. If two vectors have the same direction, what is the magnitude of the resultant? What is the direction of the resultant?

b. If two vectors of different lengths have opposite directions, what is the magnitude of the resultant? What is the direction of the resultant?

c. What is the resultant of a vector and its opposite vector?

d. What are the components of a vector with magnitude 5 units and direction east?

e. Parts c and d suggest the need for a zero vector. How could such a vector be interpreted if it represented a velocity? A displacement? A force? A translation?

Mia claimed that the magnitude of the sum of two vectors is always equal to the sum of the magnitudes of the vectors.

a. Give a counterexample to show that Mia’s claim is not true for all pairs of vectors.

b. Under what conditions is the magnitude of the sum of two vectors equal to the sum of the magnitudes of the vectors?

c. Your answers in Parts a and b are related to an important property of triangles. What is the property? Explain your reasoning.

In problems involving navigation, the direction a vector points can also be described by specifying its heading in degrees measured clockwise from the north.

a. What is the heading of a ship sailing:

i. due north? ii. due east?

ii. due south? iv. due west?

b. Sketch vectors satisfying the given criteria.

i.

_v has length 5 cm and heading 80˚.

ii.

_n has length 2 cm and heading 130˚.

c. For what angles are the direction and heading of a vector identical?

d. What relationship exists between the heading H and direction D of a vector?

In Course 3, Unit 3, Similarity and Congruence, you used properties of similar triangles to prove the Midpoint Connector Theorem:

If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side and its length is one-half the length of the third side.

a. Alonzo attempted to prove this theorem using what he learned about vectors. Check the correctness of Alonzo’s argument. Supply a reason for each statement or correct any misstep.

Suppose X and Y are the midpoints of AC and BC , respectively. Orient the vectors as shown in the diagram.

Then _{AX =}

_{XC =}

1 2 _AC

and _{CY =}

_{YB =}

1 2 _CB

_.

Also, _{AC +}

_{CB =}

_AB

and _{XC +}

_{CY =}

_XY

_.

So, ²¹ _{AC +}

¹

2 _{CB =}

_{XC +}

_CY

_or

²¹⁽_{AC +}

_{CB ) =}

_XY

_.

Therefore,

²¹ _{AB =}

_XY

_.

_XY

_||
AB

_.

It follows that ^{XY = 21AB and XY
||
AB.}

b. How, if at all, would the above vector proof need to be modified if
C in
ABC was an obtuse angle?

Quadrilaterals ABCD and AEFG are parallelograms with _BC

_{= 4AG and}

_DC

_{= 4}AE . Write a vector

proof for each of the following statements.

a. AC is a scalar multiple of

_{AF .}

b. Points A, F, and C are collinear.

Jim Chee, a helicopter pilot, wants to fly from Shiprock to Dinnebito in the Hopi-Navajo joint-use area. There is a 20-mph northwest wind.

Using the map provided in Applications Task
1, answer the following questions.

a. Suppose Jim leaves at 10:00 A.M. and travels at 100 mph heading directly for Dinnebito, with no correction for wind.

How
far is Jim from Dinnebito? What will be his location at his estimated time of arrival?

b. What course should Jim follow that accounts for the wind and ensures arriving at Dinnebito at 10:45
A.M.?

Refer to the nautical chart of the Stone Harbor, Massachusetts, region below. Suppose the Angler and Free Spirit leave the mouth of the channel at Stone Harbor together. Their directions are 55˚ and 70˚, respectively. The Angler travels at 4 knots and after 30 minutes sights the Free Spirit to the north and west. The line of sight makes an angle of 110˚ with the traveled path of the Angler from the harbor.

a. Draw the situation to scale.

b. Estimate the distance between the boats using the scale drawing.

c. Can vector component analysis be used to determine the distance between the boats? Explain your reasoning.

d. Calculate the distance between the two boats using a method other than component analysis. Compare this distance to your estimate in Part b.

e. At what average speed has the Free Spirit traveled?

In landscaping an industrial park, a large boulder was to be moved by attaching chains to two tractors that would pull at an angle of 75˚

between the chains. If one tractor can pull with 1.5 times the force of the other, and the boulder requires a force of 10,000newtons to be moved, what force is required from each tractor?

Consider the circle with radius 5 that is centered at the origin.

a. If m
AOB = 75˚, find the coordinates of point B.

b. Describe the location of another point B₁ on the circle that has the same x-coordinate as pointB. What is the relationship between the y-coordinates of the two points?

c. Describe the location of another point B₂ on the circle that has the same y-coordinate as pointB. What is the relationship between the x-coordinates of the two points?

d. Use the circle above to verify that cos 110˚ = cos 250˚.

Solve each equation or inequality.

a. (2x + 6)² = 6

b. (x – 3)(x + 8)(3x – 5) = 0 c. (2x + 1)(x – 5) = 3 d. ⁸

x = x + 7

e. (x + 4)(3 – 5x)
0

Determine the measures of the remaining sides and angle in each right triangle.

a. b.

c.

Write an equation for each line described below.

a. The line containing the points (–2, 5) and (–7, –3) b. The line parallel to 3x + y = 10 with x-intercept (10, 0) Consider the statement: If x < 0 and y < 0, then –(x + y) > 0.

a. Is the statement true or false? Provide reasoning to support your answer.

b. Write the converse of this statement.

c. Is the converse true or false? Provide reasoning to support your answer.

In 2007, 82% of New Hampshire public school seniors indicated that they planned to pursue post-secondary education. (Source: www.nhpaper.org/MAP_07_final.pdf) Suppose that you randomly chose two of those seniors.

a. What is the probability that they both indicated that they planned to pursue post- secondary
education?

b. What is the probability that neither of them planned to pursue post-secondary education?

c. What is the probability that at least one of them planned to pursue post-secondary education?

For each graph, write a function rule that produces the graph.

a. b.

c. d.

In each part, determine whether or not f(x) and g(x) are inverses of each other.

a. f(x) = 3x – 4 g(x) = ¹₃x + 4

b. f(x) = ¹_x + 2 g(x) = <sub>x


2</sub>¹ c. f(x) = 10^x

g(x) = log x

Three vertices of a parallelogram are located at P(0, 0), Q(1, 5), and R(8, 2). The fourth vertex S is also located in the first quadrant.

a. Find the coordinates of vertex S.

b. Find the lengths of each side of the parallelogram.

Rewrite each product in standard expanded form.

a. (2x – 5)² b. (7 + 3x)²

c. (12x – y)² d. (x + 1)(x² – 8x + 5)

LESSON 2 Vectors and Parametric Equations

The vectors you used in Lesson 1 were located in an east-north coordinate system. Mathematicians and many scientists use a similar system to describe the direction of a vector, namely, by using an

x-y
coordinate system in which angles are measured counterclockwise from the positive x-axis. This approach allows vectors to be analyzed using coordinate methods.

Think About This Situation

Suppose a rectangular coordinate system is placed on a nautical map so that the

Milwaukee North Shore Marina is located at the origin with the positive x-axis pointing east and the positive y-axis pointing north. A speedboat leaves the marina in a direction of 30˚

and proceeds at 16 knots.

How could the distance and direction traveled after ¹

4 hour be represented by a vector on the coordinate system? After ¹

2 hour? After 1 hour?

What are the coordinates of the boat’s position after ¹₄ hour? After ¹

2 hour? After 1 hour?

What rules would give the coordinates (x, y) of the position of the boat at any time t (in hours)?

What rules would give the coordinates (x, y) of the position of the boat at any time t (in hours) if its direction was 40˚ instead of 30˚?

In this lesson, you will learn how to analyze vectors in a coordinate system, how to use coordinate vectors to prove geometric relationships, and how to use vector components to represent linear

Coordinates and Vectors

In your previous work in Core-Plus Mathematics, you saw that there are some strong ties among algebra, geometry, and trigonometry. Flexible and coordinated use of ideas in each of those domains was often helpful in solving problems. As you work on the problems of this investigation, look for answers to this question:

What are some of the advantages of representing vectors in a standard (x, y) coordinate system?

In Lesson
1, you represented a vector

_v geometrically as an arrow and in the form

<sub>v
=
[r,

],</sub>

where r is the magnitude and
is the direction. The [6,
80˚] vector with magnitude 6 and direction 80˚ is represented on the coordinate system below.

a. Sketch the following vectors on a coordinate grid.

i. [4, 145˚] ii. [3, 240˚] iii. [5, 315˚]

b. Sketch the horizontal and vertical components of each of the three “parent” vectors in Part
a.

c. Estimate the coordinates of the terminal point of each component vector. In each case, how are the coordinates of the terminal points of the component vectors related to the coordinates of the terminal point of the given

“parent” vector? Check your conjecture in the case of the [6,
80˚] vector.

d. How can you find the coordinates (x,
y) of the terminal point of a vector if you know only its magnitude and its direction? Compare your method with those of your classmates and resolve any differences.

Suppose the speedboat in the Think About This Situation traveled at a direction of 130˚ rather than 30˚.

a. Sketch the path of the boat traveling at a speed of 16 knots on a coordinate system and identify its position at ¹₂
hour, 1
hour, and 2
hours.

b. What are the vector components of each of the positions in Part a?

c. Write rules giving the coordinates of the position of the boat for any time t hours later.

d. What rules represent the coordinates of the position of the boat for any time
t hours later and any direction
?

Vectors in Standard Position A vector with its initial point at the origin of a coordinate system is said to be in standard position and is called a position vector. Since the terminal point of every position vector has unique rectangular coordinates (x, y), the ordered pair is often identified with the vector. That is, if

_v is a position vector where the terminal point has coordinates (a,
b), then we write _v

_{= (a,
b).}

Now consider how a standard coordinate system can be used to model linear motion of an aircraft. Suppose a commercial jet leaves New York City and flies at a direction of 190˚ towards the West Coast at 600
mph. (Assume that this direction and average speed take into account the force exerted by headwinds.)

a. Model this situation by placing New York City at the origin of a coordinate system and sketch the aircraft’s path.

b. Find rules for the rectangular coordinates of the position of the aircraft t hours into the flight, assuming no deviation from the course. Then find the coordinates of the aircraft’s position after 0.25
hours and 3.2 hours.

c. What are the coordinates of the aircraft’s position when it has flown 2,000 miles?

Every vector in a coordinate system is equal to some position vector in that system.

a. Vector _u

has initial point (–10,
4) and terminal point (–2,
9) as shown at the right. What are the coordinates of the position vector that is equal to _u

? What is the [magnitude,
direction]

representation?

b. Vector

<sub>v

0</sub> has initial point (a,
b) and terminal point (c,
d). What are the coordinates of the position vector that is equal to

_{v ? What}

is the [magnitude,
direction] form?

Summarize the Mathematics

In this investigation, you explored representations of vectors in a rectangular coordinate system.

Describe the relationships among a vector, its component vectors, and the coordinates of the terminal point when the initial point of the vector is at the origin.

Suppose _OB

is a position vector, and B has coordinates (r cos
, r sin
).

i. What is the length of _OB

_?

ii. How would you draw _OB

_?

iii. Write _OB

in the [magnitude, direction] form.

Suppose a position vector _v

is represented in the [magnitude, direction] form. How can you find its coordinate representation? What does the coordinate representation tell you?

Suppose a position vector _v

is represented in rectangular coordinates (x, y). How can you find its [r,
] representation?

Be prepared to explain your responses to your classmates.

Two tugboats are maneuvering a supply barge into a Lake Superior slip. (A slip is a docking place for a boat.) One tugboat
exerts a force of 1,500 pounds with direction 340˚; another exerts a force of 2,000
pounds with direction 70˚.

a. Draw the force vectors and the resultant force as position vectors on a coordinate system with the barge at the origin.

b. Determine the coordinate forms of the three vectors. How are they related?

c. Determine the magnitude and direction of the resultant force on the barge.

Vector Algebra with Coordinates

In the previous investigation, you explored coordinate representations of vectors. In this investigation, after exploring operations on vectors in a coordinate plane, you will examine how vectors can be used to establish geometric

relationships. You will also consider the underlying vector algebra of a video game.

As you work on the following problems, look for answers to these questions:

What are some important properties of vectors and their operations?

How are these properties similar to, and different from, properties of operations with real numbers?

How can vectors and their properties be used to prove geometric statements?

Operating on Symbolic Vectors In Lesson 1, you learned about a scalar multiple, k

_{a , of}

vector a where k is any real number, and about the opposite of vector

a , denoted by –

_{a .}

a. If

a = [r,
] and 0˚

< 180˚, write each vector in [magnitude, direction] form.

i. 3

_{a ii. –}

_a

iii. –2

_{a iv. k}

a where k < 0

b. Now consider position vector

b = (4, –3). Write each scalar multiple of

b below as a position vector in coordinate form.

i. ⁵

b ii. –

b iii. ^–2

b iv. ^k

b for any real number k

c. To generalize, if position vector

a = (x, y) and k is a scalar, what are the coordinates of position vector k

a in terms of k, x, and y? Explain your reasoning.

As you have previously seen, the resultant, or sum, of two vectors can be determined geometrically by using the head-to-tail definition of a vector sum or the parallelogram law.

a. Consider position vectors

a = (2, –3) and

b = (–5, 4). Find each of the following vectors in coordinate form. It may help to use graph paper.

i.

_a+

b ii.

b + (–

_{a )}

iii. ^–2(

_{a –}

b) iv. ^–2_a

_{+ 2}

b b. General position vectors

_{a = (x}

1,y₁) and

b =(x₂,y₂) are shown at the right. Determine the coordinates of the resultant

_a+

b. Use the diagram to help explain your answer.

c. Write in words the general principle you discovered in Partb.

d. If

_{a = (x}

1,y₁), what are the coordinates of –

_{a and}

of

_{a+ (–}

_{a) ?}

e. How could you interpret (0, 0) as a vector in a motion or force situation? This special vector is called the zero vector and is sometimes written 0

.