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Chapter 8Resource Masters

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ISBN: 0-07-869135-4 Advanced Mathematical ConceptsChapter 8 Resource Masters

1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04

© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

Vocabulary Builder . . . . . . . . . . . . . . . vii-viii

Lesson 8-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 317Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 319








Chapter 8 AssessmentChapter 8 Test, Form 1A . . . . . . . . . . . . 341-342Chapter 8 Test, Form 1B . . . . . . . . . . . . 343-344Chapter 8 Test, Form 1C . . . . . . . . . . . . 345-346Chapter 8 Test, Form 2A . . . . . . . . . . . . 347-348Chapter 8 Test, Form 2B . . . . . . . . . . . . 349-350Chapter 8 Test, Form 2C . . . . . . . . . . . . 351-352Chapter 8 Extended Response

Assessment . . . . . . . . . . . . . . . . . . . . . . . 353Chapter 8 Mid-Chapter Test . . . . . . . . . . . . . 354Chapter 8 Quizzes A & B . . . . . . . . . . . . . . . 355Chapter 8 Quizzes C & D. . . . . . . . . . . . . . . 356Chapter 8 SAT and ACT Practice . . . . . 357-358Chapter 8 Cumulative Review . . . . . . . . . . . 359Unit 2 Review . . . . . . . . . . . . . . . . . . . . 361-362Unit 2 Test . . . . . . . . . . . . . . . . . . . . . . . 363-366

SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1

SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A19

Contents

© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

A Teacher’s Guide to Using theChapter 8 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 8 Resource Masters include the corematerials needed for Chapter 8. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.

All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.

Vocabulary Builder Pages vii-viii include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.

When to Use Give these pages to studentsbefore beginning Lesson 8-1. Remind them toadd definitions and examples as they completeeach lesson.

Study Guide There is one Study Guide master for each lesson.

When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent.

Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.

When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.

Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for usewith all levels of students.

When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.

© Glencoe/McGraw-Hill v Advanced Mathematical Concepts

Assessment Options

The assessment section of the Chapter 8Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessments

Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain

multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These testsare similar in format to offer comparabletesting situations.

• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.

All of the above tests include a challengingBonus question.

• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided for assessment.

Intermediate Assessment• A Mid-Chapter Test provides an option to

assess the first half of the chapter. It iscomposed of free-response questions.

• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.

Continuing Assessment• The SAT and ACT Practice offers

continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.

• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.

Answers• Page A1 is an answer sheet for the SAT and

ACT Practice questions that appear in theStudent Edition on page 549. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they mayencounter in test taking.

• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.

• Full-size answer keys are provided for theassessment options in this booklet.

© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

Chapter 8 Leveled Worksheets

Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.

• Study Guide masters provide worked-out examples as well as practiceproblems.

• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.

• Practice masters provide average-level problems for students who are moving at a regular pace.

• Enrichment masters offer students the opportunity to extend theirlearning.

primarily skillsprimarily conceptsprimarily applications

BASIC AVERAGE ADVANCED

Study Guide

Vocabulary Builder

Parent and Student Study Guide (online)

Practice

Enrichment

4

5

3

2

Five Different Options to Meet the Needs of Every Student in a Variety of Ways

1

© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.

Vocabulary Term Foundon Page Definition/Description/Example

component

cross product

direction

dot product

equal vectors

inner product

magnitude

opposite vectors

parallel vectors

parameter

(continued on the next page)

Reading to Learn MathematicsVocabulary Builder

NAME _____________________________ DATE _______________ PERIOD ________Chapter

8

© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts

Vocabulary Term Foundon Page Definition/Description/Example

parametric equation

polyhedron

resultant

scalar

scalar quantity

standard position

unit vector

vector

vector equation

zero vector

Reading to Learn MathematicsVocabulary Builder (continued)


8

© Glencoe/McGraw-Hill 317 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

8-1

Geometric VectorsThe magnitude of a vector is the length of a directed linesegment. The direction of the vector is the directed anglebetween the positive x-axis and the vector. When adding orsubtracting vectors, use either the parallelogram or thetriangle method to find the resultant.

Example 1 Use the parallelogram method to find thesum of v� and w�.Copy v� and w�, placing the initial pointstogether.Form a parallelogram that has v� and w� as twoof its sides.Draw dashed lines to represent the other twosides.

The resultant is the vector from the vertex of v�and w� to the opposite vertex of theparallelogram.

Use a ruler and protractor to measure themagnitude and direction of the resultant.

The magnitude is 6 centimeters, and thedirection is 40°.

Example 2 Use the triangle method to find 2v� � 3w�.

2v� � 3w� � 2v� � (�3w�)

Draw a vector that is twice the magnitude of v�to represent 2v�. Then draw a vector with theopposite direction to w� and three times itsmagnitude to represent �3w�. Place the initialpoint of �3w� on the terminal point of 2v�.Tip-to-tail method.

Draw the resultant from the initial point of thefirst vector to the terminal point of the secondvector. The resultant is 2v� � 3w�.


Geometric Vectors

Use a ruler and a protractor to determine the magnitude (in centimeters) and direction of each vector.

1. 2. 3.

Find the magnitude and direction of each resultant.4. x� � y� 5. x� � z�

6. 2x� � y� 7. y� � 3z�

Find the magnitude of the horizontal and vertical components of each vector shown in Exercises 1-3.

8. x� 9. y� 10. z�

11. Aviation An airplane is flying at a velocity of 500 miles per hourdue north when it encounters a wind blowing out of the west at 50 miles per hour. What is the magnitude of the airplane's resultantvelocity?

PracticeNAME _____________________________ DATE _______________ PERIOD ________

8-1

More Than Two Forces Acting on an ObjectThree or more forces may work on an object at one time. Each of theseforces can be represented by a vector. To find the resultant vector thatacts upon the object, you can add the individual vectors two at a time.

Example A force of 80 N acts on an object at anangle of 70° at the same time that aforce of 100 N acts at an angle of 150°.A third force of 120 N acts at an angleof 180°. Find the magnitude and direction of the resultant force acting onthe object.

Add two vectors at a time. The order in which the vectorsare added does not matter.

Add the 80-N vector and Now add the resulting vector the 100-N vector first. to the 120-N vector.

The resultant force is 219 N, with an amplitude of 145°.

Find the magnitude and amplitude of the resultant force acting on each object.

1. One force acts with 40 N at 50° on 2. One force acts with 75 N at 45°. Aan object. A second force acts with second force acts with 90 N at 90°.100 N at 110°. A third force acts with A third force acts with 120 N at 170°.10 N at 150°. Find the magnitude Find the magnitude and amplitude and amplitude of the resultant of the resultant force.force.


EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

8-1



8-2

Algebraic VectorsVectors can be represented algebraically using ordered pairsof real numbers.

Example 1 Write the ordered pair that represents the vectorfrom X(2, �3) to Y(�4, 2). Then find themagnitude of XY�.First represent XY� as an ordered pair.XY�� x2 � x1, y2 � y1�

� ��4 � 2, 2 � (�3)�� 6, 5�

Then determine the magnitude of XY�.

XY� � �(x�2�� x�1)�2�� (�y�2

�� y�1)�2�� (��4� �� 2�)2� �� [�2� �� (��3�)]�2�� (��6�)2� �� 5�2�� 6�1�

XY� is represented by the ordered pair ��6, 5�and has a magnitude of �6�1� units.

Example 2 Let s� � �4, 2� and t� � ��1, 3�. Find each of thefollowing.a. s� � t�

s� � t� � �4, 2� � ��1, 3�� 4 � (�1), 2 � 3�� 3, 5�

c. 4s�4s� � 4�4, 2�

� �4 � 4, 4 � 2�� 16, 8�

A unit vector in the direction of the positive x-axis is represented by i�, and a unit vector inthe direction of the positive y-axis is representedby j�. Vectors represented as ordered pairs can bewritten as the sum of unit vectors.

Example 3 Write MP� as the sum of unit vectors for M(2, 2)and P(5, 4).

First write MP� as an ordered pair.MP� � �5 � 2, 4 � 2�

� �3, 2�

Then write MP� as the sum of unit vectors.MP�� 3i� � 2j�

b. s� � t�s� � t� � �4, 2� � ��1, 3�

� �4 � (�1), 2 � 3�� 5, �1�

d. 3s� � t�3s� � t� � 3�4, 2� � ��1, 3�

� �12, 6� � ��1, 3�� 11, 9�



Algebraic Vectors

Write the ordered pair that represents AB�. Then find themagnitude of AB�.

1. A(2, 4), B(�1, 3) 2. A(4, �2), B(5, �5) 3. A(�3, �6), B(8, �1)

Find an ordered pair to represent u� in each equation if v� � �2, �1� and w� � ��3, 5�.

4. u� � 3v� 5. u� � w� � 2v�

6. u�� 4v� � 3w� 7. u� � 5w� � 3v�

Find the magnitude of each vector, and write each vector as thesum of unit vectors.

8. �2, 6� 9. �4, �5�

10. Gardening Nancy and Harry are lifting a stone statue andmoving it to a new location in their garden. Nancy is pushing thestatue with a force of 120 newtons (N) at a 60° angle with thehorizontal while Harry is pulling the statue with a force of 180 newtons at a 40° angle with the horizontal. What is the magnitude of the combined force they exert on the statue?

8-2



8-2

Basis VectorsThe expression v� � r�u� s�w, the sum of two vectors each multiplied byscalars, is called a linear combination of the vectors �uand �w.

Example Write the vector v� � � � as a linear combination of

the vectors �u� � � and �w� � �.

� � � r� � � s� � � � �–2 � 2r � s5 � 3r � 4s

Solving the system of equations yields the solution

r � – and s � – . So,�v � – �u� �w.

Write each vector as a linear combination of the vectors �u and �w.

1. �v � � �, �u� � �, �w � � � 2. �v � � �, �u� � �, �w � � �

3. �v � � �, �u� � �, �w � � � 4. �v � � �, �u� � �, �w � � �42

–1–3

2–7

1__2

104

1__2

–1

1__41

23

1–1

2–2

–34

15

16�11

3�11

16�11

3�11

2r � s 3r � 4s

1–4

23

–25

1�4

23

�25

Linear CombinationTheorem in v2

Every vector �v � v2 can be written as alinear combination of any two nonparallel vectors�uand �w. The vectors �uand �w are said to form abasis for the vector space v2 which contains all vectors having 1 column and 2 rows.



Vectors in Three-Dimensional SpaceOrdered triples, like ordered pairs, can be used to representvectors. Operations on vectors respresented by ordered triplesare similar to those on vectors represented by ordered pairs.For example, an extension of the formula for the distancebetween two points in a plane allows us to find the distancebetween two points in space.

Example 1 Locate the point at (�1, 3, 1).

Locate �1 on the x-axis, 3 on the y-axis,and 1 on the z-axis.

Now draw broken lines for parallelograms torepresent the three planes.

The planes intersect at (�1, 3, 1).

Example 2 Write the ordered triple that represents thevector from X(�4, 5, 6) to Y(�2, 6, 3). Then findthe magnitude of XY�.

XY� � (�2, 6, 3) � (�4, 5, 6)� ��2 � (�4), 6 � 5, 3 � 6�� 2, 1, �3�

�XY� � � �(x�2�� x�1)�2�� (�y�2

�� y�1)�2�� (�z2� �� z�1)�2�

� �[�2 �� (�4)]�2 � (6� � 5)2� � (3 �� 6)2�

� �(2)2 �� (1)2 �� (�3)2�� 1�4� or 3.7

Example 3 Find an ordered triple that represents 2s� � 3t� ifs� � �5, �1, 2� and t� � �4, 3, �2�.

2s� � 3t� � 2�5, �1, 2� � 3�4, 3, �2�� 10, �2, 4� � �12, 9, �6�� 22, 7, �2�

Example 4 Write AB� as the sum of unit vectors for A(5, �2, 3)and B(�4, 2, 1).

First express AB� as an ordered triple. Then writethe sum of the unit vectors i�, j�, and k�.

AB� � (�4, 2, 1) � (5, �2, 3)� ��4 � 5, 2 � (�2), 1 � 3�� 9, 4, �2�� 9i� � 4j� � 2k�

8-3


Vectors in Three-Dimensional Space

Locate point B in space. Then find the magnitude of a vector fromthe origin to B.

1. B(4, 7, 6) 2. B(4, �2, 6)

Write the ordered triple that represents AB�. Then find themagnitude of AB�.

3. A(2, 1, 3), B(�4, 5, 7) 4. A(4, 0, 6), B(7, 1, �3)

5. A(�4, 5, 8), B(7, 2, �9) 6. A(6, 8, �5), B(7, �3, 12)

Find an ordered triple to represent u� in each equation if v� � �2, �4, 5� and w� � �6, �8, 9�.

7. u� � v� � w� 8. u� � v� � w�

9. u� � 4v� � 3w� 10. u� � 5v� � 2w�

11. Physics Suppose that the force acting on an object can beexpressed by the vector �85, 35, 110�, where each measure in the ordered triple represents the force in pounds. What is themagnitude of this force?


8-3



8-3

Basis Vectors in Three-Dimensional SpaceThe expression �v � r�u� s�w � t�z, the sum of three vectors each multiplied by scalars, is called a linear combination of the vectors �u, �w,and �z.

Every vector �v � v3 can be written as a linear combination of anythree nonparallel vectors. The three nonparallel vectors, which mustbe linearly independent, are said to form a basis for v3, which containsall vectors having 1 column and 3 rows.

Example Write the vector �v � � � as a linear combination of

the vectors �u� � �, �w � � �, and �z � � �.

� �� r� �� s� �� t� �� –1 � r � s � t–4 � 3r � 2s � t

3 � r � s � t

Solving the system of equations yields the solutionr � 0, s � 1, and t � 2. So, �v ��w � 2�z.

Write each vector as a linear combination of the vectors u�, w�, and z�.

1. �v � � �, �u� � �, �w � � �, and �z � � �

2. �v � � �, �u� � �, �w � � �, and �z � � �

3. �v � � �, �u� � �, �w � � �, and �z � � �101

221

12

–1

1–12

42

–1

–101

1–23

5–20

011

101

110

–6–22

r � s � t3r � 2s � t

r � s � t

–1–11

1–21

131

–1–43

�1�1

1

1�2

1

131

�1�4

3



8-4

Perpendicular VectorsTwo vectors are perpendicular if and only if their innerproduct is zero.

Example 1 Find each inner product if u� � �5, 1�, v� � ��3, 15�,and w� � �2, �1�. Is either pair of vectors perpendicular?

a. u� � v�u� � v� � 5(�3) � 1(15)

� �15 � 15� 0

u� and v� are perpendicular.

Example 2 Find the inner product of r� and s� if r� � �3, �1, 0�and s� � �2, 6, 4�. Are r� and s� perpendicular?

r� � s� � (3)(2) � (�1)(6) � (0)(4)� 6 � (�6) � 0� 0

r� and s� are perpendicular since their innerproduct is zero.

Unlike the inner product, the cross product of two vectors isa vector. This vector does not lie in the plane of the givenvectors but is perpendicular to the plane containing the twovectors.

Example 3 Find the cross product of v� and w� if v� � �0, 4, 1�and w� � �0, 1, 3�. Verify that the resulting vectoris perpendicular to v� and w�.

v� � w� � � �� i� � � �j� � � �k� Expand by minors.

� 11i� � 0j� � 0k�� 11i� or �11, 0, 0�

Find the inner products.�11, 0, 0� � �0, 4, 1� �11, 0, 0� � �0, 1, 3)11(0) � 0(4) � 0(1) � 0 11(0) � 0(1) � 0(3) � 0

Since the inner products are zero, the crossproduct v� � w� is perpendicular to both v� and w�.

41

00

13

00

13

41

k�13

j�

41

i�00

b. v� � w�v� � w� � �3(2) � 15(�1)

� �6 �(�15)� �21

v� and w� are not perpendicular.



Perpendicular Vectors

Find each inner product and state whether the vectors areperpendicular. Write yes or no.

1. �3, 6� � ��4, 2� 2. ��1, 4� � �3, �2� 3. �2, 0� � ��1, �1�

4. ��2, 0, 1� � �3, 2, �3� 5. ��4, �1, 1� � �1, �3, 4� 6. �0, 0, 1� � �1, �2, 0�

Find each cross product. Then verify that the resulting vector isperpendicular to the given vectors.

7. �1, 3, 4� � ��1, 0, �1� 8. �3, 1, �6� � ��2, 4, 3�

9. �3, 1, 2� � �2, �3, 1� 10. �4, �1, 0� � �5, �3, �1�

11. ��6, 1, 3� � ��2, �2, 1� 12. �0, 0, 6� � �3, �2, �4�

13. Physics Janna is using a force of 100 pounds to push a cart up a ramp. The ramp is 6 feet long and is at a 30° angle with thehorizontal. How much work is Janna doing in the vertical direction? (Hint: Use the sine ratio and the formula W � F� � d�.)

8-4



8-4

Vector EquationsLet �a, �b, and �c be fixed vectors. The equation f (x) � �a � 2x�b � x2�cdefines a vector function of x. For the values of x shown, theassigned vectors are given below.

If �a � �0, 1�, �b � �1, 1�, and �c � �2, –2�, the resulting vectors for thevalues of x are as follows.

For each of the following, complete the table of resulting vectors.

1. f (x) � x3�a � 2x2�b � 3x�c�a � �1, 1� �b � �2, 3� �c � �3, –1�

2. f (x) � 2x2�a � 3x �b � 5�c�a � �0, 1, 1� �b � �1, 0, 1� �c � �1, 1, 0�

3. f (x) � x2�c � 3x�a � 4 �b �a � �1, 1, 1� �b � �3, 2, 1� �c � �0, 1, 2�

4. f (x) � x3�a � x �b � 3�c�a � �0,1, –2� �b � �1, –2, 0� �c � �–2, 0, 1�

x –2 –1 0 1 2

f (x) �a � 4�b � 4 �c �a � 2 �b � �c �a �a � 2 �b � �c �a � 4 �b � 4 �c

x –2 –1 0 1 2

f (x) �12,–3� �4, 1� �0, 1� �0,–3� �4,–11�

x f (x)

–1012

x f (x)

–2–101

x f (x)

0123

x f (x)

–1013



8-5

Applications with VectorsVectors can be used to represent any quantity that hasdirection and magnitude, such as force, velocity, and weight.

Example Suppose Jamal and Mike pull on the ends of arope tied to a dinghy. Jamal pulls with a force of 60 newtons and Mike pulls with a force of 50 newtons. The angle formed when Jamal andMike pull on the rope is 60°.

a. Draw a labeled diagram that representsthe forces.

Let F�1 and F�2 represent the two forces.

b. Determine the magnitude of theresultant force.First find the horizontal (x) and vertical ( y)components of each force.Given that we place F�1 on the x-axis, the unitvector is 1i� � 0j�.Therefore, the x- and y-components of F�1 are60i� � 0j�.F�2 � xi� � yj�

cos 60° � �5x0�

x � 50 cos 60°� 25

Thus, F�2 � 25i� � 43.3j�.

Then add the unit components.

(60i� � 0j�) � (25i� � 43.3j�) � 85i� � 43.3j�

F� �8�5�2�� 4�3�.3�2� �9�0�9�9�.8�9� 95.39

The magnitude of the resultant force is 95.39 newtons.

c. Determine the direction of the resultant force.tan � � �48

35.3� Use the tangent ratio.

� � tan�1 �4835.3�

� 27°

The direction of the resultant force is 27° withrespect to the vector on the x-axis.

sin 60° � �5y0�

y � 50 sin 60° 43.3


Applications with Vectors

Make a sketch to show the given vectors.1. a force of 97 newtons acting on an object while a force of 38 newtons

acts on the same object at an angle of 70° with the first force

2. a force of 85 pounds due north and a force of 100 pounds due westacting on the same object

Find the magnitude and direction of the resultant vector for eachdiagram.3. 4.

5. What would be the force required to push a 200-pound object up aramp inclined at 30° with the ground?

6. Nadia is pulling a tarp along level ground with a force of 25pounds directed along the tarp. If the tarp makes an angle of 50°with the ground, find the horizontal and vertical components ofthe force.

7. Aviation A pilot flies a plane east for 200 kilometers, then 60°south of east for 80 kilometers. Find the plane's distance anddirection from the starting point.


8-5


NAME _____________________________ DATE _______________ PERIOD ________

Enrichment8-5

Linearly Dependent VectorsThe zero vector is �0, 0� in two dimensions, and �0, 0, 0� in threedimensions.

A set of vectors is called linearly dependent if and only if thereexist scalars, not all zero, such that a linear combination of the vectors yields a zero vector.

Example Are the vectors �–1, 2, 1�, �1, –1, 2�, and �0, –2, –6�linearly dependent?

Solve a�–1, 2, 1� � b�1, –1, 2� � c�0, –2, –6� � �0, 0, 0�.–a � b � 0

2a � b � 2c � 0a � 2b � 6c � 0

The above system does not have a unique solution. Anysolution must satisfy the conditions that a � b � 2c.

Hence, one solution is a � 1, b � 1, and c � .

�–1, 2, 1� � �1, –1, 2� � �0, –2, –6� � �0, 0, 0�, so thethree vectors are linearly dependent.

Determine whether the given vectors are linearly dependent. Write yes or no. If theanswer is yes, give a linear combination that yields a zero vector.

1. �–2, 6�, �1, –3� 2. �3, 6�, �2, 4�

3. �1, 1, 1�, �–1, 0, 1�, �1, –1, –1� 4. �1, 1, 1�, �–1, 0, 1�, �–3, –2, –1�

5. �2, –4, 6�, �3, –1, 2�, �–6, 8, 10� 6. �1, –2, 0�, �2, 0, 3�, �–1, 1, �9�4

1�2


Vectors and Parametric Equations

Vector equations and parametric equations allow us tomodel movement.

Example 1 Write a vector equation describing a line passingthrough P1(8, 4) and parallel to a� � �6, �1�. Thenwrite parametric equations of the line.

Let the line � through P1(8, 4) be parallel to a�.For any point P2(x, y) on �, P1P2�x � 8, y � 4�.Since P1P2 is on � and is parallel to a�, P1P2 � ta�,for some value t. By substitution, we have �x � 8, y � 4� � t�6, �1�.

Therefore, the equation �x � 8, y � 4� � t�6, �1�is a vector equation describing all of the points (x, y) on � parallel to a� through P1(8, 4).

Use the general form of the parametricequations of a line with �a1, a2� � �6, �1�and �x1, y1� � �8, 4�.

x � x1 � ta1 y � y1 � ta2x � 8 � t(6) y � 4 � t(�1)x � 8 � 6t y � 4 � t

Parametric equations for the line are x � 8 � 6tand y � 4 � t.

Example 2 Write an equation in slope-intercept form of theline whose parametric equations are x � �3 � 4tand y � 3 � 4t.

Solve each parametric equation for t.

x � �3 � 4t y � 3 � 4tx � 3 � 4t y � 3 � 4t�x �

43� � t �

y �

43

� � t

Use substitution to write an equation for the linewithout the variable t.

�x �4

3� � �y �

43

� Substitute.(x � 3)(4) � 4( y � 3) Cross multiply.

4x � 12 � 4y � 12 Simplify.y � x � 6 Solve for y.


8-6



Vectors and Parametric Equations

Write a vector equation of the line that passes through point Pand is parallel to a�. Then write parametric equations of the line.

1. P(�2, 1), a� � �3, �4� 2. P(3, 7), a� � �4, 5�

3. P(2, �4), a� � �1, 3� 4. P(5, �8), a� � �9, 2�

Write parametric equations of the line with the given equation.5. y � 3x � 8 6. y � �x � 4

7. 3x � 2y � 6 8. 5x � 4y � 20

Write an equation in slope-intercept form of the line with thegiven parametric equations.

9. x � 2t � 3 10. x � t � 5y � t � 4 y � �3t

11. Physical Education Brett and Chad are playing touch footballin gym class. Brett has to tag Chad before he reaches a 20-yardmarker. Chad follows a path defined by �x � 1, y � 19� � t�0, 1�,and Brett follows a path defined by �x � 12, y � 0� � t��11, 19�.Write parametric equations for the paths of Brett and Chad. WillBrett tag Chad before he reaches the 20-yard marker?

8-6



8-6

Using Parametric Equations to Find theDistance from a Point to a PlaneYou can use parametric equations to help you find the distance from apoint not on a plane to a given plane.

Example Find the distance from P(�1, 1, 0) to the plane x � 2y � z � 4.

Use the coefficients of the equation of the plane and thecoordinates of the point to write the ratios below.

� �

The denominators of these ratios represent a vector thatis perpendicular to the plane, and passes through thegiven point.

Set t equal to each of the above ratios. Then, t = ,

t = , and t = .

So, x � t �1, y � 2t � 1, and z � –t are parametric equations of the line.

Substitute these values into the equation of the plane.(t � 1) � 2(2t � 1) � (–t) � 4Solve for t: 6t � 1 � 4

t �

This means that t � at the point of

intersection of the vector and the plane.

The point of intersection is � � 1, 2� � � 1, � �

Use the distance formula:

d � ��1�� 12��2�� (1� �� 2�)2�� 0� �� 12��2� 1.2 units

Find the distance from the given point to the given plane. Round your answers to thenearest tenth.

1. from (2, 0, –1) to x � 2y � z � 3 2. from (1, 1, –1) to 2x � y � 3z � 5

1�2

1�2

1�2

1�2

1�2

z � 0�

–1y � 1�

2

x + 1�

1

z � 0�

–1y � 1�

2x � 1�

1

or �� , 2, � �.1�2

1�2



Modeling Motion Using Parametric EquationsWe can use the horizontal and vertical components of aprojectile to find parametric equations that represent thepath of the projectile.

Example 1 Find the initial horizontal and vertical velocitiesof a soccer ball kicked with an initial velocity of33 feet per second at an angle of 29° with theground.

v�x � v� cos � v�y � v� sin �v�x � 33 cos 29° v�y � 33 sin 29°v�x 29 v�y 16

The initial horizontal velocity is about 29 feetper second and the initial vertical velocity isabout 16 feet per second.

The path of a projectile launched from the ground may bedescribed by the parametric equations x � tv� cos � for horizontal distance and y � tv� sin � � �12�gt2 for vertical distance, where t is time and g is acceleration due to gravity.Use g 9.8 m/s2 or 32 ft/s2.

Example 2 A rock is tossed at an intitial velocity of 50 meters per second at an angle of 8° with the ground. After 0.8 second, how far has the rock traveled horizontally and vertically?

First write the position of the rock as a pair ofparametric equations defining the postition ofthe rock for any time t in seconds.

x � tv� cos � y � tv� sin � � �12�gt2

x � t(50) cos 8° y � t(50) sin 8° � �12�(9.8)t2 v� � 50 m/s

x � 50t cos 8° y � 50t sin 8° � 4.9t2

Then find x and y when t � 0.8 second.

x � 50(0.8) cos 8° y � 50(0.8) sin 8° � 4.9(0.8)2

39.61 2.43

After 0.8 second, the rock has traveled about 39.61 meters horizontally and is about 2.43 metersabove the ground.

8-7


Modeling Motion Using Parametric Equations

Find the initial horizontal and vertical velocity for each situation.1. a soccer ball kicked with an initial velocity of 39 feet per second at

an angle of 44° with the ground

2. a toy rocket launched from level ground with an initial velocity of63 feet per second at an angle of 84° with the horizontal

3. a football thrown at a velocity of 10 yards per second at an angleof 58° with the ground

4. a golf ball hit with an initial velocity of 102 feet per second at anangle of 67° with the horizontal

5. Model Rocketry Manuel launches a toy rocket from groundlevel with an initial velocity of 80 feet per second at an angle of80° with the horizontal.a. Write parametric equations to represent the path of the rocket.

b. How long will it take the rocket to travel 10 feet horizontallyfrom its starting point? What will be its vertical distance atthat point?

6. Sports Jessica throws a javelin from a height of 5 feet with aninitial velocity of 65 feet per second at an angle of 45° with theground.a. Write parametric equations to represent the path of the

javelin.

b. After 0.5 seconds, how far has the javelin traveled horizontallyand vertically?


8-7



8-7

Coordinate Equations of ProjectilesThe path of a projectile after it is launched is a parabola when graphedon a coordinate plane.

The path assumes that gravity is the only force acting on the projectile.

The equation of the path of a projectile on the coordinate plane is given by,

y � –� � x2 � (tan �)x,

where g is the acceleration due to gravity, 9.8 m/s2 or 32 ft/s2,v0 is the initial velocity, and � is the angle at which the projectile is fired.

Example Write the equation of a projectile fired at an angleof 10° to the horizontal with an initial velocity of120 m/s.

y � –� � x2 � (tan 10°)x

y � –0.00035x2 � 0.18x

Find the equation of the path of each projectile.

1. a projectile fired at 80° to the 2. a projectile fired at 40° to thehorizontal with an initial velocity horizontal with an initial velocity of 200 ft/s of 150 m/s

9.8��2(120)2 cos2 10°

g��2v0

2 cos2�


Transformation Matrices in Three-Dimensional Space

Example 1 Find the coordinates of the vertices of thepyramid and represent them as a vertex matrix.

A(�2, �2, �2)B(2, �2, �2)C(2, 2, �2)D(�2, 2, �2)E(0, 0, 2)

The vertex matrix for the pyramid is � .

Example 2 Let M represent the vertex matrix of the pyramid in Example 1.

a. Find TM if T � � �.b. Graph the resulting image and describe the

transformation represented by matrix T.

a. First find TM.

TM � � � � � � b. Then graph the points

represented by the resulting matrix.

The transformation matrix reflects the imageof the pyramid over the xz-plane.

E002

D�2�2�2

C2

�2�2

B22

�2

A�2

2�2

002

�22

�2

22

�2

2�2�2

�2�2�2

001

0�1

0

100

001

0�1

0

100

E002

D�2

2�2

C22

�2

B2

�2�2

A�2�2�2

xyz


8-8

� � � � �



Transformation Matrices in Three-Dimensional Space

Write the matrix for each figure.1. 2.

Translate the figure in Question 1 using the given vectors. Graph each image andwrite the translated matrix.3. a� �1, 2, 0� 4. b� ��1, 2, �2�

Transform the figure in Question 2 using each matrix. Graph eachimage and describe the result.5.

� 6.

� 00

�1

010

100

002

020

200

8-8



8-8

Spherical CoordinatesThere are many coordinate systems for locating a point in thetwo-dimensional plane. You have studied one of the most com-mon systems, rectangular coordinates. The most commonlyused three-dimensional coordinate systems are the extendedrectangular system, with an added z-axis, and the sphericalcoordinate system, a modification of polar coordinates.

Note that the orientation of the axes shown is a different perspective than that used in your textbook.

Point P(d, �, �) in three-dimensional space is located usingthree spherical coordinates:

d � distance from origin� � angle relative to x -axis� � angle relative to y-axis

The figure at the right shows point Q with rectangular coor-dinates (2, 5, 6).

1. Find OA and AB.

2. Find OB by using the Pythagorean theorem.

3. Find QB.

4. Find d.

5. Use inverse trigonometric functions to find � and � to the nearestdegree. Write the spherical coordinates of Q.

Find the spherical coordinates of the point with the given rectangular coordinates.Round distances to the nearest tenth and angles to the nearest degree.

6. (4, 12, 3)

7. (–2, –3, –1)

8. (a, b, c)



8 Chapter 8 Test, Form 1A

Write the letter for the correct answer in the blank at the right of each problem.

1. The vector v� has a magnitude of 89.7 feet and a direction of 12° 48�. 1. ________Find the magnitude of its vertical component.A. 887.47 ft B. 19.87 ft C. 19.38 ft D. 87.58 ft

2. What is an expression for x� involving r�, s�, and t�? 2. ________A. �3r� � s� � t� B. �3r� � s� � t�C. 3r� � s� � t� D. 3r� � s� � t�

3. Find the ordered pair that represents the vector from 3. ________A(�4.3, �0.9) to B(�2.8, 0.2). Then find the magnitude of AB�.A. �1.5, 1.1�; 3.46 B. ��7.1, �0.7�; 7.13C. �1.5, 1.1�; 1.86 D. ��7.1, �1.1�; 7.18

4. Find the ordered triple that represents the vector from A(�1.4, 0.3, �7.2) 4. ________to B(0.4, �9.1, 8.2). Then find the magnitude of AB�.A. �1.8, �9.4, 15.4�; 18.13 B. ��1, �8.8, 1�; 8.91C. �1.8, �9.4, 15.4�; 12.33 D. ��1, �8.8, 1�; 8.80

5. Find an ordered pair to represent u� in u� � �34� w�� 2v� if w�� 23�, 4� 5. ________and v� � ��38�, �2�.A. ��14�, 7� B. ��54�, �1� C. ��14�, 4� D. ��54�, 7�

6. Find an ordered triple to represent x� in x� � �6 z� � �14�y� if y� � �2, 18, ��45�� 6. ________and z� � ��12�, �34�, ��16��.A. ��28

5�, 0, �45�� B. ��72�, 0, �45�� C. ��72�, 0, �65�� D. ��72�, �383�, �45��

7. Write MN� as the sum of unit vectors for M��34�, 5, �23�� and N�6, �9, �35��. 7. ________

A. �247� i� � 14 j� � �1

15� k� B. �24

1� i� � 14 j� � �115� k�

C. 9i� � �8290� j� � �94

3� k� D. �247� i� � 14 j� � �1

15� k�

8. Find the inner product of a� and bb� if a� � �4, �54�, ��13�� and 8. ________bb� � ��12�, �2, ��32��, and state whether the vectors are perpendicular.A. 5; no B. 5; yes C. 0; yes D. 0; no

9. Find the cross product of v� and w� if v� � ��13�, 4, ��38�� and w�� 6, ��45�, 4�. 9. ________

A. ��11507�, ��11

12�, ��31

556�� B. ��11

603�, �11

12�, ��31

556��

C. ��11507�, 3, ��31

556�� D. ��11

603�, ��11

12�, ��31

556��

10. Find the magnitude and direction of the 10. ________resultant vector for the diagram at the right.A. 8.2 N, 73° 35�B. 20 N, 18° 37�C. 6.5 N, 79° 7� D. 8.2 N, 83° 48�

11. A force F�1 of 35 newtons pulls at an angle of 15° north of due east. 11. ________A force F�2 of 75 newtons pulls at an angle of 55° west of due south.Find the magnitude and direction of the resultant force.A. 43.8 N, 54.1° west of due south B. 43.8 N, 39.1° west of due southC. 42.2 N, 54.1° west of due south D. 42.2 N, 27.4° west of due south


Write a vector equation of the line that passes through point P and is parallel to a�. Then write parametric equations of the line.

12. P(�1, 3); a� � ��6, �1� 12. ________A. �x � 1, y � 3� � t��6, �1�; x � �1 � 6t, y � 3 � tB. �x � 1, y � 3� � t��6, �1�; x � 1 � 6t, y � �3 � tC. �x � 1, y � 3� � t��6, �1�; x � �1 � 6t, y � 3 � tD. �x � 1, y � 3� � t��6, �1�; x � 1 � 6t, y � �3 � t

13. P(0, 5); a� � �2, �9� 13. ________A. �x, y � 5� � t��2, 9�; x � �2t, y � 5 � 9tB. �x, y � 5� � t�2, �9�; x � 2t, y � 5 � 9tC. �x � 2, y �9� � t�0.5�; x � 2, y � �9 � 5tD. �x � 2, y � 9� � t�0, �5�; x � 2, y � �9 � 5t

14. Which graph represents a line whose parametric equations are 14. ________x � 2t � 4 and y � �t � 2?A. B. C. D.

15. Write parametric equations of �3x � �12�y � �23�. 15. ________

A. x � t; y � 6t � �43� B. x � t; y � 6t � �31�

C. x � t; y � 6t � �13� D. x � t; y � 6t � �34�

16. Write an equation in slope-intercept form of the line whose 16. ________parametric equations are x � ��12�t � �23� and y � t � �34�.

A. y � 2x � �172� B. y � 2x � �1

72� C. y � �2x ��1

72�D. y � �2x � �1

72�

Darius serves a volleyball with an initial velocity of 34 feet persecond 4 feet above the ground at an angle of 35°.17. What is the maximum height, reached after about 0.61 seconds? 17. ________

A. 2.14 ft B. 9.94 ft C. 5.94 ft D. 6.14 ft18. After how many seconds will the ball hit the ground if it landed 39 feet 18. ________

away and it is not to be returned?A. 1.2 B. 1.3 C. 1.4 D. 0.4

A triangular prism has vertices at A(2, �1, �1), B(2, 1, 4), C(2, 2, �1), D(�1, �1, �1), E(�1, 1, 4), and F(�1, 2, �1).19. Which image point has the coordinates (�3, 2, 1) after a translation 19. ________

using the vector ��5, 1, �3�?A. C� B. B� C. E� D. F�

20. What point represents a reflection of B over the yz-plane? 20. ________A. B�(�2, �1, 4) B. B�(�2, 1, 4)C. B�(�2, 2, �4) D. B�(�2, 1, �4)

Bonus Find the cross product of ��34� v� and �12� w� if v� � ��2, 12, �3� Bonus: ________and w� � ��7, 4, �6�.

A. ��425�, ��28

7�, ��527��B. ��62

3�, ��287�, ��52

7�� C. ��425�, �28

7�, ��527�� D. ��42

5�, ��287�, ��62

9��

Chapter 8 Test, Form 1A (continued)


8


Chapter 8 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. The vector v� has a magnitude of 6.1 inches and a direction of 55°. Find 1. ________the magnitude of its vertical component.A. 5.00 in. B. 10.64 in. C. 7.45 in. D. 3.50 in.

2. What is an expression for x� involving r� and s�? 2. ________A. r� � 2s� B. �r� � 2s�C. r� � 2s� D. �r� � 2s�

3. Find the ordered pair that represents the vector from A(9, 2) to 3. ________B(�6, 3). Then find the magnitude of AB�.A. ��15, 1�; 15.03 B. �3, 5�; 5.83C. �15, �1�; 3.74 D. �3, 1�; 3.16

4. Find the ordered triple that represents the vector from A(�3, 5, 6) to 4. ________B(�6, 8, 6). Then find the magnitude of AB�.A. �3, �3, 0�; 4.24 B. ��9, 13, 12�; 19.85C. ��3, 3, 0�; 4.24 D. ��9, 3, 0�; 9.49

5. Find an ordered pair to represent u� in u� � 4w� � 2v� if w� � ��3, 4� 5. ________and v� � ��4, 0�.A. ��20, 16� B. ��4, 16� C. ��10, �8� D. ��22, 8�

6. Find an ordered triple to represent x� in x� � 3z� � 5y� if y� � �2, 11, �5� 6. ________and z� � ��2, 8, 6�.A. �4, 79, �7� B. ��16, �31, 43�C. ��2, 17, �1� D. �16, �7, �45�

7. Write MN� as the sum of unit vectors for M(�14, 8, 6) and N(7, 9, �2). 7. ________A. �7i� � j� � 8k� B. �7i� � j� � 8k�

C. 21i� � j� � 8k� D. 21i� � j� � 8k�

8. Find the inner product of a� and b� if a� � �4, �2, �2� and b� � ��7, �2, 4� 8. ________and state whether the vectors are perpendicular.A. 0; yes B. �32; yes C. �40; no D. �32; no

9. Find the cross product of v� and w� if v� � ��9, 4, �8� and w� � �6, �2, 4�. 9. ________A. ��54, �8, �32� B. �0, �12, �6�C. �32, 84, 42� D. ��6, �12, 0�

10. Find the magnitude and direction of the 10. ________resultant vector for the diagram at the right.A. 26.4 N; 51.8° B. 22.2 N; 58.8°C. 22.2 N; 38.8° D. 26.4 N; 31.8°

11. An 18-newton force acting at 56° and a 32-newton force acting at 124° 11. ________act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?A. 42.2 N; 100.7° B. 42.2 N; 280.7°C. 44.6 N; 36.5° D. 44.6 N; 216.5°

Chapter

8


Write a vector equation of the line that passes through point Pand is parallel to a�. Then write parametric equations of the line.12. P(�2, 5); a� � ��7, �6� 12. ________

A. �x � 2, y � 5� � t��7, �6�; x � �2 � 7t, y � 5 � 6tB. �x � 2, y � 5� � t��7, �6�; x � 2 � 7t, y � �5 � 6tC. �x � 2, y � 5� � t��7, �6�; x � 2 � 7t, y � �5 � 6tD. �x � 2, y � 5� � t��7, �6�; x � �2 � 7t, y � 5 � 6t

13. P(0, 3); a� � �1, �8� 13. ________A. �x, y � 3� � t��1, 8�; x � �t, y � 3 � 8tB. �x � 1, y � 8� � t(0, 3); x � 1, y � �8 � 3tC. �x, y � 3� � t�1, �8�; x � t, y � 3 � 8tD. �x � 1, y � 8� � t�0, �3�; x � 1, y � �8 � 3t

14. Which is the graph of parametric equations x � 4t � 5 and y � �4t � 5? 14. ________A. B. C. D.

15. Write parametric equations of x � 4y � 5. 15. ________A. x � t; y � �4t � �54� B. x � t; y � ��14�t � �54�

C. x � t; y � 4t � �54� D. x � t; y � �14�t � �54�

16. Write an equation in slope-intercept form of the line whose 16. ________parametric equations are x � �3t � 8 and y � �2t � 9.A. y � �23�x � �43

3� B. y � ��23�x � �433� C. y � ��23�x � �13

1� D. y � �23�x � �131�

Aaron kicked a soccer ball with an initial velocity of 39 feet persecond at an angle of 44° with the horizontal.17. After 0.9 second, how far has the ball traveled horizontally? 17. ________

A. 24.4 ft B. 12.3 ft C. 11.4 ft D. 25.2 ft

18. After 1.5 seconds, how far has the ball traveled vertically? 18. ________A. 6.1 ft B. 40.6 ft C. 4.6 ft D. 42.1 ft

A triangular prism has vertices at A(2, �1, 0), B(2, 1, 0), C(2, 0, 2), D(�1, �1, 0), E(�1, 1, 0), and F(�1, 0, 2).19. Which image point has the coordinates (�2, 1, 1) after a translation 19. ________

using the vector ��1, 2, 1�?A. C′ B. D′ C. E′ D. F′

20. What point represents a reflection of E over the xz-plane? 20. ________A. E′(1, �1, 0) B. E′(�1, �1, 0)C. E′(�1, 1, 0) D. E′(2, �1, 0)

Bonus Find 3v� � �2w� if v� � ��1, 5, 3� and w� � ��7, 5, �6�. Bonus: ________A. �270, 162, �180� B. �270, 90, 240�C. �270, �90, 240� D. �270, �162, �180�

Chapter 8 Test, Form 1B (continued)


8


Chapter 8 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.1. The vector v� has a magnitude of 5 inches and a direction of 32°. 1. ________

Find the magnitude of its vertical component.A. 4.24 in B. 2.65 in C. 2.79 in D. 31.88 in

2. What is an expression for x� involving r� and s� ? 2. ________A. �r� � s� B. r� � s�C. �r� � s� D. r� � s�

3. Find the ordered pair that represents the vector from 3. ________A(1, 2) to B(0, 3). Then find the magnitude of AB�.A. ��1, 1�; 1.41 B. �1, �1�; 2C. ��1, �1�; 1.41 D. �1, 1�; 2

4. Find the ordered triple that represents the vector from A(�4, 2, 1) to 4. ________B(�3, 0, 5). Then find the magnitude of AB�.A. ��7, �2, 4�; 8.31 B. ��1, �2, 4�; 4.58C. �1, �2, 4�; 4.58 D. ��7, 2, 6�; 9.43

5. Find an ordered pair to represent u� in u� � 2w� � v� if w� � ��2, 4� and 5. ________v� � �3, 1�.A. ��7, �7� B. ��1, �7� C. ��7, 7� D. ��1, 7�

6. Find an ordered triple to represent x� in x� � 3 y� � z� if y� � �2, �1, 5� 6. ________and z� � �1, �6, 6�.A. �7, 3, 9� B. �5, 3, 9� C. �5, 9, 9� D. �7, 3, 21�

7. Write MN� as the sum of unit vectors for M(�2, 3, 6) and N(1, 5, �2). 7. ________A. �i� � 2 j� � 8k� B. �i� � 2 j� � 4k�

C. 3i� � 2 j� � 4k� D. 3i� � 2 j� � 8k�

8. Find the inner product of a� and b� if a� � �3, 0, �1� and b� � �4, �2, 5� and 8. ________state whether the vectors are perpendicular.A. 7; no B. 0; yes C. 7; yes D. 0; no

9. Find the cross product of v� and w� if v� � ��1, 2, 4� and w� � ��3, �1, 5�. 9. ________A. �14, �7, �5� B. �14, 7, 7� C. �14, �7, 7� D. �6, �7, 7�

10. Find the magnitude and direction of the resultant 10. ________vector for the diagram at the right.A. 129.5 N, 46.5° B. 129.5 N, 11.5°C. 113.6 N, 13.1° D. 113.6 N, 48.1°

11. A 22-newton force acting at 48° and a 65-newton 11. ________force acting at 24° act concurrently on an object.What is the magnitude and direction of a third force that produces equilibrium on the object?A. 85.6 N; 30° B. 85.6 N; 6°C. 85.6 N; 210° D. 85.6 N; 186°

Chapter

8


Write a vector equation of the line that passes through point P and is parallel to a� . Then write parametric equations of the line.12. P(�1, 3); a� � �2, �5� 12. ________

A. �x � 1, y � 3� � t�2, �5�; x � 1 � 2t, y � �3 � 2tB. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � 3 � 5tC. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � 3 � 5tD. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � �3 � 2t

13. P(1, �4); a� � �2, �5� 13. ________A. �x � 2, y � 5� � t�1, �4�; x � 2 � t, y � �5 � 4tB. �x � 2, y � 5� � t�1, �4�; x � �2 � t, y � 5 � 4tC. �x � 1, y � 4� � t�2, �5�; x � 1 � 2t, y � �4 � 5tD. �x � 1, y � 4� � t�2, �5�; x � �1 � 2t, y � 4 � 5t

14. Which graph represents a line whose parametric equations are 14. ________x � t � 2 and y � t � 2?A. B. C. D.

15. Write parametric equations of y � 2x � 3. 15. ________A. x � t; y � �12� t � 3 B. x � t; y � 2t � 3

C. x � t; y � �12� t � 3 D. x � t; y � 2t � 316. Write an equation in slope-intercept form of the line whose 16. ________

parametric equations are x � t � 4 and y � 2t � 1.A. y � 2x � 7 B. y � 2x � 9 C. y � 2x � 5 D. y � �12� x � 5

Jana hit a golf ball with an initial velocity of 102 feet per second at an angle of 67° with the horizontal.17. After 2 seconds, how far has the ball traveled horizontally? 17. ________

A. 27.9 ft B. 123.8 ft C. 79.7 ft D. 97.7 ft18. After 3 seconds, how far has the ball traveled vertically? 18. ________

A. 137.7 ft B. 119.6 ft C. 233.7 ft D. 52.6 ft

A triangular prism has vertices at A(2, 0, 0), B(2, 1, 3), C(2, 2, 0), D(0, 0, 0), E(0, 1, 3), and F(0, 2, 0).19. Which image point has the coordinates (1, 4, 3) after a translation 19. ________

using the vector �1, 2, 3�?A. C� B. D� C. E� D. F�

20. What point represents a reflection of B over the xy-plane? 20. ________A. B�(2, 1, �3) B. B�(�2, �1, 3)C. B�(�2, 1, �3) D. B�(2, �1, 3)

Bonus Find the cross product of v� and �2w� if v� � �2, 4, �1� and Bonus: ________w� � ��1, 2, �5�.

A. �44, 22, �16� B. �36, �22, �16� C. �36, 22, �16� D. �36, �22, 0�

Chapter 8 Test, Form 1C (continued)


8


Chapter 8 Test, Form 2A

NAME _____________________________ DATE _______________ PERIOD ________

1. The vector v� has a magnitude of 11.4 meters and a direction 1. __________________of 248°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 89.6 inches. If v� � ��72� u�, 2. __________________what is the magnitude of v�?

Use a ruler and a protractor to determine the magnitude (in centimeters) and direction of each vector. Then find the magnitude and direction of each resultant.

3. �3 a� � �12� b� � �23� a� 3. __________________

4. �12� a� � �25� b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(1.8, �3.8) to B(�0.1, 5.1). Then find the magnitude of AB�.

6. A force F�1 of 18.8 newtons pulls at an angle of 12° above 6. __________________due east. A force F�2 of 3.2 newtons pulls at an angle of 88° below due east. Find the magnitude and direction of the resultant force.

Find an ordered pair or ordered triple to represent u� in each equation if v� � �0, �1

2��, w� � �2, ��3

4��, r� � �1, ��1

4�, 2�, 7. __________________

and s� � �10, �6, �34

��. 8. __________________

7. u� � �v� � �13� w� 8. u� � �12� r� � 4s� 9. u� � ��23� s� � 3r� 9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(5.1, �0.8, 9) to B(�3.8, 7, �1.4). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(2.1, �2.6, 7) 11. __________________and F(�0.8, �7, 5).

Find each inner product and state whether the vectors are 12. __________________perpendicular. Write yes or no.

12. �8, �23�� 12�, �6� 13. ��2, 6, 8� � ��4, �2, ��12�� 13. __________________

Find each cross product. 14. __________________

14. �6, ��12�, 3� � �4, 2, ��13�� 15. ��14�, 7, �4� � ��5, �32�, 2� 15. __________________

Chapter

8


16. Find the magnitude and direction of the 16. __________________resultant vector for the diagram at the right.

17. What force is required to push a 147-pound crate up 17. __________________a ramp that makes a 12° angle with the ground?

18. A 12.2-newton force acting at 12° and an 18.9-newton force 18. __________________acting at 75.8° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P ��32�, �5� and is parallel to a� � �2, 3�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation. 20. __________________

20. y � ��34�x � 3 21. 2x � �13�y � 5 21. __________________

Write an equation in slope-intercept form of the line with the given parametric equations. 22. __________________

22. x � ��12�t � 6; y � 2t � 4 23. x � �2t � 5; y � 4t � �47� 23. _______________________________________________

24. Lisset throws a softball from a height of 4 meters, with 24. __________________an initial velocity of 20 meters per second at an angle of 45° with respect to the horizontal. When will the ball be a horizontal distance of 30 meters from Lisset?

25. A rectangular prism has vertices at A(1, �1, 3), B(1, 2, 3), 25. __________________C(1, 2, �1), D(1, �1, �1), E(�2, �1, 3), F(�2, 2, 3),G(�2, 2, �1), and H(�2, �1, �1). Find the vertices of the prism after a translation using the vector �1, �2, 1�and then a reflection over the xy-plane.

Bonus Write parametric equations for the line passing Bonus: __________________

through the point at ��23�, ��34�� and perpendicular

to the line with equation 4y � 8x � 3.

Chapter 8 Test, Form 2A (continued)


8


Chapter 8 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________

1. The vector v� has a magnitude of 10 meters and a direction 1. __________________of 92°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 25.5 feet. If v� � ��13�u�, what 2. __________________is the magnitude of v�?


3. a� � 3b� 3. __________________

4. ��12�a� � b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(0, �8) to B(�1, 7). Then find the magnitude of AB�.

6. A force F�1 of 27 newtons pulls at an angle of 23° above 6. __________________due east. A force F�2 of 33 newtons pulls at an angle of 55°below due west. Find the magnitude and direction of the resultant force. 7. __________________

Find an ordered pair or ordered triple to represent u� ineach equation if v� � �1, �6�, w� � �2, �5�, r� � �1, �1, 0�, and 8. __________________s� � �10, �6, 5�.

7. u� � v� � 3w� 8. u� � 3s� �2r� 9. u� � r� � �15�s� 9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(5, �8, 9) to B(�2, 2, 2). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(1, �2, 7) and 11. __________________F(�8, �7, 5).

Find each inner product and state whether the vectors are perpendicular. Write yes or no.12. �8, 2� � �0, �6� 12. __________________

13. �3, �7, 4� � ��4, �2, 1� 13. __________________

Find each cross product.14. �6, �4, 3� � �4, 2, �6� 14. __________________

15. ��2, 7, �4� � ��5, �6, 2� 15. __________________

Chapter

8


16. Find the magnitude and direction of the resultant vector 16. __________________for the diagram below.

17. Anita is riding a toboggan down a hill. If Anita weighs 17. __________________120 pounds and the hill is inclined at an angle of 72°from level ground, what is the force that propels Anita down the hill?

18. A 15-newton force acting at 30° and a 25-newton force 18. __________________acting at 60° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P(1, 0) and is parallel to a� � ��3, �7�. Then write parametric equations of the line and graph it.

20. __________________Write parametric equations for each equation.20. y � �x �3 21. 2x � 4y � 5 21. __________________

Write an equation in slope-intercept form of the line 22. __________________with the given parametric equations.22. x � �t � 6; y � 2t � 4 23. x ��2t � 5; y � 4t � 2 23. __________________

24. Pablo kicks a football with an initial velocity of 30 feet 24. __________________per second at an angle of 58° with the horizontal. After 0.3 second, how far does the ball travel vertically?

25. A rectangular prism has vertices at A(2, 0, 2), B(2, 2, 2),C(2, 2, �2), D(2, 0, �2), E(0, 0, 2), F(0, 2, 2), G(0, 2, �2), and H(0, 0, �2). Find the vertices of the prism after a reflection over the xz-plane. 25. __________________

Bonus Write parametric equations for the line passing Bonus: __________________through (2, �2) and parallel to the line with equation 8x � 2y � �6.

Chapter 8 Test, Form 2B (continued)


8


Chapter 8 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________

1. The vector v� has a magnitude of 5 meters and a direction 1. __________________of 60°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 4 centimeters. 2. __________________If v� � ��3

1� u�, what is the magnitude of v� ?

Use a ruler and protractor to determine the magnitude (in centimeters) and direction of each vector. Then find the magnitude and direction of each resultant.

3. a� � b� 3. ____________

4. 2a� � b� 4. ____________

5. Write the ordered pair that represents the vector from 5. __________________A(3, �1) to B(�1, �2). Then find the magnitude of AB�.

6. A force F�1 of 25 newtons pulls at an angle of 20° above 6. __________________due east. A force F�2 of 35 newtons pulls at an angle of 60°above due east. Find the magnitude and direction of the resultant force.

Find an ordered pair or ordered triple to represent u� in each equation if v� � �2, �3�, w� � �1, 5�, r� � �1, �1, 1�, 7. __________________and s� � �0, �3, 2�.

8. __________________7. u� � �2v� � w� 8. u� � s� � r� 9. u� � 3s� � r�

9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(1, 3, 5) to B(�3, 0, 1). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(5, 1, �4) and 11. __________________F(9, 3, 1).

Find each inner product and state whether the vectors are perpendicular. Write yes or no. 12. __________________

12. �2, 0� � �0, �5� 13. �3, �4, �2� � ��2, �2, 1� 13. __________________


14. �2, �1, 3� � �1, 0, �5� 15. ��2, 2, �1� � �0, �2, 2� 15. __________________

Chapter

8


16. Find the magnitude and direction 16. __________________of the resultant vector for the diagram at the right.

17. Matt is pushing a grocery cart on a level floor with a force 17. __________________of 15 newtons. If Matt’s arms make an angle of 28° with the horizontal, what are the vertical and horizontal components of the force?

18. A 10-newton force acting at 45° and a 20-newton force 18. __________________acting at 130° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P(�3, 2) and is parallel to a� � ��2, 6�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation.

20. y � 4x 21. y � 2x � 1 20. __________________

21. __________________

Write an equation in slope-intercept form of the line with the given parametric equations.

22. x � t; y � 2t 23. x � 2t; y � t � 5 22. __________________

23. __________________

24. Shannon kicks a soccer ball with an initial velocity of 24. __________________45 feet per second at an angle of 12° with the horizontal.After 0.5 second, what is the height of the ball?

25. A cube has vertices at A(2, 0, 0), B(2, 0, 2), C(2, 2, 2), 25. __________________D(2, 2, 0), E(0, 0, 0), F(0, 0, 2), G(0, 2, 2), and H(0, 2, 0).Find the vertices of the prism after a translation using the vector �1, �1, 2�.

Bonus Write parametric equations for the line passing Bonus: __________________through (0, 0) and parallel to 3y � 9x � 3.

Chapter 8 Test, Form 2C (continued)


8

x

y

O


Chapter 8 Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. Given the vectors below, complete the questions that follow.

c� � ��3, 1�, and d� � ��8, �11�

a. Show two ways to find a� � b�.

b. Find a� � b�. Explain each step.

c. Does a� � b� � b� � a�? Why or why not?

d. Does a� � b� � b� � a�? Defend your answer.

e. Tell how to find the sum c� � d�. Find the sum and its magnitude.

f. Find two vectors whose difference is �4, �1, 3�. Write the difference as the sum of unit vectors.

g. Find a vector perpendicular to �7, �3�. Explain how you know that the two vectors are perpendicular.

h. Find a� � b� if a� � �2, 1, 0� and b� � �1, 3, 0�. Graph the vectors and the cross product c� in three dimensions.

2. a. Find parametric equations for a line parallel to a� � �3, �1� and passing through (�2, 4).

b. Find another vector and point from which the parametric equations for the same line can be written.

3. A ball is thrown with an initial velocity of 56 feet per second at an angle of 30° with the ground.

a. If the ball is thrown from 8 feet above ground, when will it hit the ground?

b. How far will the ball travel horizontally before hitting the ground?

4. Find two pairs of perpendicular vectors. Then verify that they are perpendicular by calculating their dot products.

Chapter

8


Chapter 8 Mid-Chapter Test (Lessons 8-1 through 8-4)


81. The vector v� has a magnitude of 12 inches and direction 1. __________________

of 36°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 9.9 centimeters. If 2. __________________v� � �4u�, what is the magnitude of v�?


3. 2a� � 2b�

3. ____________

4. �3a� � b�

4. ____________

5. Write the ordered pair that represents the vector from 5. __________________A(4, �7) to B(0, �5). Then find the magnitude of AB�.

6. Write CD� as the sum of unit vectors for points C(4, �3) 6. __________________and D(1, �2).

7. Javier normally swims 3 miles per hour in still water. When 7. __________________he tries to swim directly toward shore at the beach, his course is altered by the incoming tide. If the current is 6 mph and makes an angle of 47� with the direct path to shore, what is Javier’s resultant speed?

Find an ordered pair to represent u� in each equation if 8. __________________v� � ��3, 8� and w� � �3, �4�. 9. __________________

8. u� � �5w� 9. u� � 2v� � 3w� 10. u� � 4w� � v� 10. __________________11. Write the ordered triple that represents the vector from 11. __________________

A(2, �2, 4) to B(6, 1, �8). Then find the magnitude of AB�.

12. Write EF� as a sum of unit vectors for E(1, �4, 3) and 12. __________________F(�4, �2, 3).

Find an ordered triple to represent u� in each equation if 13. __________________v� � �5, �2, 0� , w� � �3, �8, 1� , and x� � �0, �3, �4�. 14. __________________

13. u� � �v� � w� 14. u� � 3w� � 2x� 15. u� � x� � �12� v� 15. __________________

Find each inner product and state whether the vectors are perpendicular. Write yes or no. 16. __________________

16. �6, �4� � �2, 4� 17. �4, �3, 1� � �8, 12, 4� 17. __________________


18. �9, 1, 0� � ��3, 2, 5� 19. �6, �4, �2� � �1, 1, �3� 19. __________________

20. Find a vector that is perpendicular to both c� � �0, �3, 6� 20. __________________and d� � �4, 2, �5�.

1. The vector v� has a magnitude of 13 millimeters and a 1. __________________direction of 84°. Find the magnitude of its vertical and horizontal components.

2. The vector a� has a magnitude of 6.3 meters. If b� � �2a� , 2. __________________what is the magnitude of b�?


3. 2a� � b� 3. __________________

4. �a� � 2b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(1, �3) to B(�6, �8). Then find the magnitude of AB�.

6. Write CD� as a sum of unit vectors for C(7, �4) and D(�8, 1). 6. __________________

7. Two people are holding a box. One person exerts a force of 7. __________________140 pounds at an angle of 65.5� with the horizontal. The other person exerts a force of 115 pounds at an angle of 58.3� with the horizontal. Find the net weight of the box.

Find an ordered pair to represent u� in each equation if 8. __________________

v� � �6, �6� and w� � �3, �4�. 9. __________________

8. u� � �5w� 9. u� � 2v� � 3w� 10. u� � 4w� � v� 10. _____________

1. Write the ordered triple that represents the vector from 1. __________________A(3, 4, 10) to B(8, 4, �2). Then find the magnitude of AB�.

2. Write EF� as a sum of unit vectors for E(8, 2, �4) and 2. __________________F(5, �3, 0).

3. Find an ordered triple that represents 2v� � �31�w� � z� if 3. __________________

v� � �2, �1, 5�, w� � ��3, 4, �6�, and z� � �0, 3, �2�.

4. Find the inner product of a� and b� if a� � �7, �3, 8� and 4. __________________b� � �5, �2, �4�. Are a� and b� perpendicular?

5. Find the cross product of c� and d� if c� � �5, �5, 4� and 5. __________________d� � �2, 3, �6�. Verify that the resulting vector is perpendicular to c� and d�.

Chapter 8, Quiz B (Lessons 8-3 and 8-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 8, Quiz A (Lessons 8-1 and 8-2)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

8

Chapter

8

1. Find the magnitude and direction 1. __________________of the resultant vector forthe figure at the right.

2. Maggie is pulling on a tarp along level ground with a force of 2. __________________25 newtons. If the tarp makes an angle of 50� with the ground,what are the vertical and horizontal components of the force?

3. A 25-newton force acting at 75� and a 50-newton force acting at 3. __________________45° act concurrently on an object. What are the magnitude and direction of a third force that produces equilibrium on the object?

4. Write a vector equation of the line that passes through 4. __________________P(1, �3) and is parallel to q� � ��2, 4�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation.5. 6x � y � 2 6. �2x � 5y � �4 5. __________________

6. __________________

Write an equation in slope-intercept form of the line with the given parametric equations. 7. __________________

7. x � 6t � 8 8. x � 3t � 10y � �t � 4 y � �4t � 2 8. __________________

While positioned 25 yards directly in front of the goalposts, Bill kicks the football withan initial velocity of 65 feet per second at an angle of 35� with the ground.

1. Write the position of the football as a pair of parametric 1. __________________equations. If the crossbar is 10 feet above the ground, does Bill’s team score?

2. What is the elapsed time from the moment the football is 2. __________________kicked to the time the ball hits the ground?

A rectangular prism has vertices at A(�1, �1, 1), B(�1, 1, 1), C(�1, 1, �2), D(�1, �1, �2), E(2, �1, 1), F(2, 1, 1), G(2, 1, �2), and H(2, �1, �2). Find the vertices of the prism after each transformation.

3. a translation using the vector �1, 2, �1� 3. __________________

4. a reflection over the yz-plane 4. __________________

5. the dimensions are increased by a factor of 3 5. __________________

Chapter 8, Quiz D (Lessons 8-7 and 8-8)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 8, Quiz C (Lessons 8-5 and 8-6)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

8

Chapter

8

x

y

O


Chapter 8 SAT and ACT Practice

NAME _____________________________ DATE _______________ PERIOD ________

After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. If the area of a circle is 49�, what

is the circumference of the circle?A 7B 7�C 14D 14�E 49

2. If all angles in the figure below areright angles, find the area of the shaded region.A 12 units2

B 48 units2

C 144 units2

D 192 units2

E 240 units2

3. What is the equation of the perpendicular bisector of the segment from P(2, �1) to Q(3, 7)?A 2x � 16y � 53B 2x � 16y � 53C 2x � 16y � 43D 2x � 16y � 43E None of these

4. If A(0, 0) and B(8, 4) are vertices of�ABC and �ABC is isosceles, whatare the coordinates of C?A (5, �9)B (8, �3)C (5, 5)D (8, 0)E (1, 8)

5. The following are the dimensions offive rectangular solids. All have thesame volume EXCEPTA 8 by 6 by 5B 4 by 15 by 2C �15� by 15 by 40

D �13� by 24 by 15

E �12� by 4 by 60

6. In �ABC, �A is a right angle. If BC � 25 and AB � 20, which is thearea of �ABC?A 187.5 units2

B 250 units2

C 75�3�4� units2

D 150 units2

E 300 units2

7. If the measure of one angle in a parallelogram is 40°, what are the measures of the other three angles?A 60°, 100°, and 160°B 40°, 280°, and 280°C 40°, 140°, and 140°D 40°, 150°, and 150°E None of these

8. Which of the following statements isNOT true for the diagram below?

A m�6 � m�9B m�3 � m�6 � 90°C m�2 � m�6 � m�5 � 180°D m�8 � m�2 � m�3E m�4 � m�2 � m�9

9. If 2 y � 50 and y � 2x � 1, then which of the following statements is true?A x � 13B 16.5 x 32.5C 2 x 2.5D 3 x 3.5E None of these

10. If x and y are real numbers and y2 � 6 � 2x, then which of the following statements is true?A x 6B x 3C x � 6D x � 3E None of these

Chapter

8


11. A circle is inscribed in a square asshown in the figure below. What is theratio of the area of the shaded regionto the area of the square?A �4

��

B �1 �4

��

C �4 �4

��

D ��4�

E �1 �4

��

12. Each angle in the figure below is aright angle. Find the perimeter of thefigure.A 11 unitsB 18 unitsC 22 unitsD 24 unitsE 28 units

13. Which number is �45� of �34� of 10?A 6 B 4C 3 D 1.5E 0.5

14. Evaluate 9[4�2(�2)4 � 3�2]�1.A 8 B �8

1�

C ��81� D �8

E None of these

15. A solid cube has 4-inch sides. Howmany straight cuts through the cubeare needed to produce 512 small cubesthat have half-inch sides?A 7 B 9C 16 D 21E None of these

16. A roll of wallpaper is 15 inches wideand can cover 39 square feet. How longis the roll?A 2.6 ft. B 21.7 ft.C 31.2 ft D 46.9 ft.E None of these

17–18. Quantitative ComparisonA if the quantity in Column A is

greaterB if the quantity in Column B is

greaterC if the two quantities are equalD if the relationship cannot be

determined from the informationgiven

Column A Column B

17. Square X has sides of length x.Square Y has sides of length 2x.

18. ABCD is a rectangle.

19. Grid-In �BDE is contained in rectangle ABCD as shown below. Findthe area of �BDE in square units.

20. Grid-In The area of a rhombus is 28 square units. The length of one diagonal is 7 units. What is the lengthof the other diagonal in units?

Chapter 8 SAT and ACT Practice (continued)


8

A D

CB E

A

D C

BE

9

3

4

4

7 Area of square XHalf the area of square Y

Area of �DBC Area of �AED


Chapter 8 Cumulative Review (Chapters 1-8)

NAME _____________________________ DATE _______________ PERIOD ________

1. Find the zero of ƒ(x) � 12 � 4x. If no zero exists, write none. 1. __________________

2. Graph ƒ(x) � �x � 3�. 2.

3. Triangle ABC has vertices A(�2, 3), B(2, 1), and C(0, �4). 3. __________________Find the image of the triangle after a reflection over the x-axis.

4. Find the inverse of � if it exists. If it does not 4. __________________

exist, write none.

5. Write the equation obtained when ƒ(x) � �x� is 5. __________________translated 3 units down and compressed horizontally by a factor of 0.5.

6. Solve �x � 3� � 5. 6. __________________

7. Determine the rational roots of 2x3 � 3x2 �17x � 12 � 0. 7. __________________

8. Solve �x �1

1� � �2x� 0. 8. __________________

9. Identify all angles that are coterminal with a 232� angle. 9. __________________Then find one positive angle and one negative angle coterminal with the given angle.

10. Find the area of �ABC if a � 4.2, A � 36�, and B � 55°. 10. __________________

11. Find the amplitude and period of y � 3 cos �4x�. 11. __________________

12. Find the phase shift of y � 2 sin �x � ��6��. 12. __________________

13. If � is an angle in the second quadrant and cos � � � ��35��, 13. __________________

find tan 2�.

14. Write 2x � y � 5 in normal form. Then find the length 14. __________________of the normal and the angle it makes with the positive x-axis.

15. Write an equation in slope-intercept form of the line 15. __________________whose parametric equations are x � �3 � 7t and y � 4 � t.

�10

�35

Chapter

8

Blank


Unit 2 Review, Chapters 5-8

NAME _____________________________ DATE _______________ PERIOD ________

Find the value of the given trigonometricfunction for angle � in standard positionif a point with the given coordinates lieson its terminal side.

1. cos �; (2, 3) 2. tan �; (10, 2)3. sin �; (�4, 1) 4. sec �; (1, 0)

Solve each problem. Round to thenearest tenth.

5. If A � 25° and a � 12.1, find b.6. If a � 3 and B � 59° 2’, find c.7. If c � 24 and B � 63°, find a.

Evaluate each expression.

8. cos �Arccos �14��9. cot �Cos�1 �23��

10. cos (Sin�1 0) � sin (Tan�1 0)

Determine the number of possiblesolutions for each triangle. If a solutionexists, solve the triangle. Round to thenearest tenth.11. A � 46°, a � 86, c � 20012. a � 19; b � 20, A � 65°13. A � 73°; B � 65°, b � 38

Find the area of each triangle. Round tothe nearest tenth.14. a � 5, b � 9, c � 615. a � 22, A � 63°, B � 17°

Change each radian measure to degreemeasure.

16. ��2� 17. �34��

18. �72�� 19. ��1

72��

Solve.20. Given a central angle of 60°, find the

length of its intercepted arc in a circleof radius 6 inches. Round to thenearest tenth.

Find each value by referring to the graphof the sine or the cosine function.

21. sin � 22. cos �2��

23. sin �72�� 24. cos (�6�)

State the amplitude and period for eachfunction.25. y � 2 cos 3x26. y � �5 tan 5x27. y � 4 cot ��2

x� � �2��

Graph each function.

28. y � �12� cos 2x

29. y � 3 tan �2x � �2��

30. y � x � 2 sin 3x

Write the equation for the inverse ofeach relation. Then graph the relationand its inverse.31. y � arccos x 32. y � cot x

Use the given information to determineeach trigonometric value.

33. sec � � �43�, 0° � 90°; cos �

34. cos � � �13�, 0° � 90°; sin �

35. sin � � �13�, 0° � 90°; cot �

Verify that each equation is an identity.36. tan x � tan x cot2 x � sec x csc x37. sin (180° � �) � tan � cos �

Use sum or difference identities to findthe exact value of each trigonometricfunction.38. sin 105° 39. cos 135°40. tan 15° 41. sin (�210°)

UNIT2


If x is an angle in the first quadrant andsin x � �2

5�, find each value.

42. cos 2x 43. sin �2x�

44. tan �2x� 45. sin 2x

Solve each equation for 0° � x � 180°.46. sin2 x � sin x � 047. cos 2x � 4 cos x � 348. 5 cos x � 1 � 3 cos 2x

Write each equation in normal form.Then find the length of the normal andthe angle that it makes with the positive x-axis.49. 2x � 3y � 2 � 050. 5x � �2y � 851. y � 3x � 7

Find the distance between the point withthe given coordinates and the line withthe given equation. 52. (2, 5); 2x � 2y � 3 � 053. (�2, 2); �x � 4y � �654. (1, �3); 4x � y � 1 � 0

Use vectors a� and b� for Exercises 55-56.

55. Use a ruler and a protractor to determine the magnitude (in centimeters) and direction of theresultant a� � b�.

56. Find the magnitude of the vertical andhorizontal components of a�.

Find an ordered pair to represent a� ineach equation if b � �1, �3� and c� � �2, �2�.57. a� � b� � c� 58. a� � b� � c�59. a� � 3b� � 2c� 60. a� � �3b� � c�

Find an ordered triple to represent u� ineach equation if v� � �3, 1, �1� and w� � ��5, 2, 3� . Then write u� as the sumof unit vectors.61. u� � 2v� � w� 62. u� � v� � 2w�63. u� � 3v� � 3w� 64. u� � 4v� � 2w�

Find each inner product or crossproduct.65. �4, �2� � ��2, 3�66. �3, �4, 1� � �4, �2, 2�67. �5, �2, 5� � ��1, 0, �3�

Write a vector equation of the line thatpasses through point P and is parallel to v�. Then write parametric equations ofthe line.68. P(0, 5), v� � ��1, 5�69. P(4, �3), v� � ��2, �2�

Unit 2 Review, Chapters 5-8 (continued)

NAME _____________________________ DATE _______________ PERIOD ________UNIT2


Unit 2 Test, Chapters 5-8

NAME _____________________________ DATE _______________ PERIOD ________

1. True or false: sin (�85°) � �sin 85°. 1. __________________

2. Find the area of �ABC if a � 12, b � 15, and c � 23. Round 2. __________________to the nearest square unit.

3. Write the equation 5x � y � 2 � 0 in normal form. 3. __________________

4. Graph the function y � 2 cos �� 3��. 4.

5. Given a central angle of 60°, find the length of its 5. __________________intercepted arc in a circle of radius 15 inches. Round to the nearest tenth.

6. A vector has a magnitude of 18.3 centimeters and a direction 6. __________________of 38°. Find the magnitude of its vertical and horizontal components to the nearest tenth.

7. Write parametric equations of y � 5x � 2. 7. __________________

8. Find the value of Sin�1 �sin �56��. 8. __________________

9. Use the Law of Sines to solve �ABC when a � 1.43, 9. __________________b � 4.21, and A � 30.4°. If no solution exists, write none.

10. Use the sum or difference identity to find the exact value 10. __________________of tan 105°.

11. Find the distance between P(7, �4) and the line with 11. __________________equation x � 3y � 5 � 0. Round to the nearest tenth.

12. Find the inner product of the vectors �2, 5� and �4, �2�. 12. __________________Then state whether the vectors are perpendicular.Write yes or no.

UNIT2


13. Find the value of sin � for angle � in standard position if a 13. __________________point with coordinates (�3, 2) lies on its terminal side.

14. Solve sin � � �1 for all real values of �. 14. __________________

15. A car’s f lywheel has a timing mark on its outer edge. 15.The height of the timing mark on the rotating flywheel is given by y � 3.55 sin �x � �

�4��. Graph one full cycle

of this function.

16. Find the ordered pair that represents �3 w� if w� � �6, �4�. 16. __________________

17. Write XY� as the sum of unit vectors for X(8, 2, �9) and 17. __________________Y(�12, �1, 10).

18. In the triangle at the right, b � 6.2 18. __________________and c � 8.2. Find � to the nearest tenth.

19. If 0° � 90° and tan � � ��23�� , f ind cos �. 19. __________________

20. Solve sin2 x � sin x � 2 � 0 for 0° � x 360°. 20. __________________

21. If �849° is in standard position, determine a coterminal 21. __________________angle that is between 0° and 360°. State the quadrant in which the terminal side lies.

22. Verify that �tansexccxsc x� � 1 is an identity. Write your 22. __________________

answer on a separate piece of paper.

Unit 2 Test, Chapters 5-8 (continued)




NAME _____________________________ DATE _______________ PERIOD ________

23. Find the cross product of the vectors �2, �1, 4� and �6, �2, 1�. 23. __________________Is the resulting vector perpendicular to the given vectors?

24. A triangular shelf is to be placed in a curio cabinet whose 24. __________________sides meet at an angle of 105°. If the edges of the shelf along the sides measure 56 centimeters and 65 centimeters, how long is the outside edge of the shelf ? Round to the nearest tenth.

25. If sin � � �35� and � is a second quadrant angle, find tan 2�. 25. __________________

26. Graph the function y � sin x on 26.the interval ��2

�� x � �2��.

27. Change �79�� radians to degree measure. 27. __________________

28. Nathaniel pulls a sled along level ground with a force of 28. __________________30 newtons on the rope attached to the sled. If the rope makes an angle of 20° with the ground when it is pulled taut, find the horizontal and vertical components of the force. Round to the nearest tenth.

29. State the amplitude, period, and phase shift of the 29. __________________function y � �2 sin (4� � 2�).

30. If � and � are two angles in Quadrant II such that 30. __________________

tan � � ��12� and tan � � ��23�, find cos (� � � ).

31. A surveyor sets a stake and then walks 150 feet north, 31. __________________where she sets a second stake. She then walks 300 feet east and sets a third stake. How far from the first stake is the third stake? Round to the nearest tenth.

UNIT2


32. Find the value of Tan�1 ��13��. 32. __________________

33. Use the Law of Cosines to solve � ABC with A � 126.3°, 33. __________________b � 45, and c � 62.5. Round to the nearest tenth.

34. Write an equation in slope-intercept form of the line with 34. __________________parametric equations x � 2 � 3t and y � 4 � t.

35. Verify that cos (90° � A) � �sin A is an identity. 35. __________________

36. Write the equation for the inverse of the function 36. __________________y � Cos x. Then graph the function and its inverse.

37. Find sin (Sin�1 �14�). 37. __________________

38. Find the area of a sector if the central angle measures 38. __________________

�56�� radians and the radius of the circle is 8 centimeters.

Round to the nearest tenth.

39. Find the measure of the reference angle for 400°. 39. __________________

40. A golf ball is hit with an initial velocity of 135 feet per 40. __________________second at an angle of 22° above the horizontal. Will the ball clear a 25-foot-wide sand trap whose nearest edge is 300 feet from the golfer?



© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(10 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

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SAT and ACT Practice Answer Sheet(20 Questions)

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w��

2v�

�6,�

3��

7, 7

�

6.u�

�4v�

�3w�

7.u�

�5w�

�3v�

��1,

11�

��21

, 28�

Fin

d t

he

mag

nit

ud

e of

eac

h v

ecto

r, an

d w

rite

eac

h v

ecto

r as

th

esu

m o

f u

nit

vec

tors

.8.

�2, 6

�9.

�4,�

5�

2�1�0�

; 2 i�

�6

j��

4�1�; 4

i��

5 j�

10.G

ard

enin

gN

ancy

an

d H

arry

are

lift

ing

a st

one

stat

ue

and

mov

ing

it t

o a

new

loca

tion

in t

hei

r ga

rden

. Nan

cy is

pu

shin

g th

est

atu

e w

ith

a f

orce

of

120

new

ton

s (N

) at

a 6

0°an

gle

wit

h t

he

hor

izon

tal w

hil

e H

arry

is p

ull

ing

the

stat

ue

wit

h a

for

ce o

f 18

0 n

ewto

ns

at a

40°

angl

e w

ith

th

e h

oriz

onta

l. W

hat

is t

he

mag

nit

ude

of

the

com

bin

ed f

orce

th

ey e

xert

on

th

e st

atu

e?29

5.62

N

8-2

© G

lenc

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w-H

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2A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

8-2

Ba

sis

Vec

tors

Th

e ex

pres

sion

v��

r�u�

s�w,

the

sum

of

two

vect

ors

each

mu

ltip

lied

by

scal

ars,

is c

alle

d a

lin

ear

com

bin

atio

nof

th

e ve

ctor

s �u

and �w

.

Exa

mp

leW

rite

th

e ve

ctor

v��

��a

s a

lin

ear

com

bin

atio

n o

f

the

vect

ors

�u�

��an

d�w

��

�.

��

r��

�s �

��

�– 2

�2r

�s

5�

3r�

4s

Sol

vin

g th

e sy

stem

of

equ

atio

ns

yiel

ds t

he

solu

tion

r�

–an

d s

�–

. S

o,�v

� –

�u�

�w.

Wri

te e

ach

vec

tor

as

a lin

ear

com

bin

atio

n o

f th

e ve

ctor

s�u a

nd

�w.

1.�v

� ��

, �u�

��,�w

� �

�2.

�v�

��, �u

��

,�w

� �

��v

�6�u

��w

�v�

�u�

4�w

3.�v

� �

�,�u�

��,�w

� �

�4.

�v�

��,�u

� �

�,�w�

��v

�–

�u�

��w�v

��u

��w

13 � 1016 � 5

1 � 2

4 2– 1 – 3

2 – 7

1 __ 2 10 4

1 __ 2 – 1

19 � 2

1 __ 4 12 3

1 – 12 – 2

– 3 41 5

16 � 113 � 11

16 � 113

� 11

2r�

s

3r�

4s1 – 4

2 3– 2 5

1�

42 3

�2 5

Lin

ear

Com

bin

atio

nT

heo

rem

in v

2

Eve

ry v

ecto

r�v

� v

2ca

n be

writ

ten

as a

linea

r co

mbi

natio

n of

any

two

nonp

aral

lel v

ecto

rs�u

and�w

. T

he v

ecto

rs�u

and �w

are

said

to fo

rm a

basi

s fo

r th

e ve

ctor

spa

cev 2

whi

ch c

onta

ins

all v

ecto

rs h

avin

g 1

colu

mn

and

2 ro

ws.



© G

lenc

oe/M

cGra

w-H

ill32

4A

dva

nced

Mat

hem

atic

al C

once

pts

Vec

tors

in

Th

ree

-Dim

en

sio

na

l S

pa

ce

Loca

te p

oin

t B

in s

pac

e. T

hen

fin

d t

he

mag

nit

ud

e of

a v

ecto

r fr

omth

e or

igin

to

B.

1.B

(4, 7

, 6)

2.B

(4,�

2, 6

)

�1�0�

1�2

�1�4�

Wri

te t

he

ord

ered

tri

ple

th

at r

epre

sen

ts A

B�

. Th

en f

ind

th

em

agn

itu

de

of A

B�

.

3.A

(2, 1

, 3),

B(�

4, 5

, 7)

4.A

(4, 0

, 6),

B(7

, 1,�

3)

��6,

4, 4

�; 2�

1�7��3

, 1,�

9�; �

9�1�

5.A

(�4,

5, 8

), B

(7, 2

,�9)

6.A

(6, 8

,�5)

, B(7

,�3,

12)

�11,

�3,

�17

�; �

4�1�9�

�1,�

11, 1

7�; �

4�1�1�

Fin

d a

n o

rder

ed t

rip

le t

o re

pre

sen

t u�

in e

ach

eq

uat

ion

if

v��

�2,�

4, 5

�an

d w�

��6

,�8,

9�.

7.u�

�v�

�w�

8.u�

�v�

�w�

�8,�

12, 1

4��

4, 4

,�4�

9.u�

�4v�

�3w�

10.

u��

5v��

2w�

�26,

�40

, 47�

��2,

�4,

7�

11.P

hys

ics

Su

ppos

e th

at t

he

forc

e ac

tin

g on

an

obj

ect

can

be

expr

esse

d by

th

e ve

ctor

�85,

35,

110

�, w

her

e ea

ch m

easu

re in

th

e or

dere

d tr

iple

rep

rese

nts

th

e fo

rce

in p

oun

ds. W

hat

is t

he

mag

nit

ude

of

this

for

ce?

�14

3 lb

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

8-3

© G

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5A

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Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

8-3

Ba

sis

Vec

tors

in T

hre

e-D

ime

nsi

on

al S

pa

ce

Th

e ex

pres

sion

�v�

r�u�

s�w�

t�z, t

he

sum

of

thre

e ve

ctor

s ea

ch

mul

tipl

ied

by s

cala

rs, i

s ca

lled

a li

nea

r co

mb

inat

ion

of t

he v

ecto

rs�u, �w

,an

d�z.

Eve

ry v

ecto

r �v

� v

3ca

n b

e w

ritt

en a

s a

lin

ear

com

bin

atio

n o

f an

yth

ree

non

para

llel

vec

tors

. T

he

thre

e n

onpa

rall

el v

ecto

rs, w

hic

h m

ust

be li

nea

rly

inde

pen

den

t, a

re s

aid

to f

orm

a b

asis

for

v 3, w

hic

h c

onta

ins

all v

ecto

rs h

avin

g 1

colu

mn

an

d 3

row

s.

Exa

mp

le

Wri

te t

he

vect

or�v

��

�as a

lin

ear

com

bin

atio

n o

f

the

vect

ors

�u��

,�w

��

,an

d�z

��

.

��

r ��

s ��

t ��

�–

1�

r�

s�

t–

4�

3r�

2s�

t3

�r

�s

�t

Sol

vin

g th

e sy

stem

of

equ

atio

ns

yie

lds

the

solu

tion

r�

0, s

�1,

an

d t

�2.

So,

�v��w

�2�z

.

Wri

te e

ach

vec

tor

as a

lin

ear

com

bin

atio

n o

f th

e ve

ctor

s u�

, w�, a

nd

z�.

1.�v

� ��

, �u

� ��

,�w

� ��

,an

d �z

� ��

�v�

–5�u

� �w

�3�z

2.�v

� ��

, �u�

��,�w

� ��

,an

d �z

� ��

�v�

�u�

�w �

�z

3.�v

� ��

, �u

� ��

,�w

� ��

,an

d �z

� ��

�v�

–�u

��z

3 � 21 � 2

1 0 1

2 2 1

1 2 – 1

1 – 1 2

1 � 723 � 7

8 � 7

4 2 – 1

– 1 0 1

1 – 2 3

5 – 2 0

0 1 1

1 0 1

1 1 0

– 6 – 2 2

r�

s�

t3r

�2s

�t

r�

s�

t

– 1 – 1 1

1 – 2 1

1 3 1

– 1 – 4 3

�1

�1 1

1�

2 1

1 3 1

�1

�4 3

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Mat

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once

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Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

8-4

Vec

tor

Eq

ua

tio

ns

Let

�a, � b

, an

d�c

be f

ixed

vec

tors

. Th

e eq

uat

ion

f(x

)��a

�2x

� b�

x2�c

defi

nes

a v

ecto

r fu

nct

ion

of

x. F

or t

he

valu

es o

fx

show

n, t

he

assi

gned

vec

tors

are

giv

en b

elow

.

If �a

��0

, 1�,

� b �

�1, 1

�, an

d �c

��2

, –2�

, th

e re

sult

ing

vect

ors

for

the

valu

es o

f x

are

as f

ollo

ws.

For

each

of

the

follo

win

g, c

omp

lete

th

e ta

ble

of

resu

ltin

g v

ecto

rs.

1.f(

x)�

x3 �a�

2x2� b

�3x

�c�a

��1

, 1�

� b �

�2, 3

��c

��3

, –1�

2.f(

x)�

2x2 �a

�3x

� b �

5�c�a

��0

, 1, 1

�� b

��1

, 0, 1

��c

��1

, 1, 0

�

3.f(

x)�

x2�c

�3x

�a�

4� b

�a

��1

, 1, 1

�� b

��3

, 2, 1

��c

��0

, 1, 2

�

4.f(

x)�

x3 �a�

x� b

�3�c

�a�

�0,1

, –2�

� b �

�1, –

2, 0

��c

��–

2, 0

, 1�

x–2

–10

12

f(x)

�a�

4� b

�4

�c�a

�2

� b�

�c�a

�a�

2� b

� �c

�a�

4� b

�4

�c

x–2

–10

12

f(x)

�12,

–3�

�4, 1

��0

, 1�

�0,–

3��4

,–11

�

xf(

x)

–1�–

14,–

4�0

�0, 0

�1

�6, –

8�2

�10,

–22

�

xf(

x)

–2�–

11, 3

, 2�

–1�–

8, –

3,–1

�0

�–5,

–5,

0�

1�–

2, –

3, 5

�

xf(

x)

0�–

12, –

8, –

4�1

�–9,

–4,

1�

2�–

6, 2

, 10�

3�–

3, 1

0, 2

3�

xf(

x)

–1�–

5,–3

,5�

0�–

6, 0

, 3�

1�–

7, 3

, 1�

3�–

9, 3

3, –

51�

© G

lenc

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7A

dva

nced

Mat

hem

atic

al C

once

pts

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Pe

rpe

nd

icu

lar

Vec

tors

Fin

d e

ach

inn

er p

rod

uct

an

d s

tate

wh

eth

er t

he

vect

ors

are

per

pen

dic

ula

r. W

rite

yes

or

no.

1.�3

, 6��

��4,

2�

2.��

1, 4

��3

,�2�

3.�2

, 0��

��1,

�1�

0; y

es�

11; n

o�

2; n

o

4.��

2, 0

, 1��

�3, 2

,�3�

5.��

4,�

1, 1

��1

,�3,

4�

6.�0

, 0, 1

��1

,�2,

0�

�9;

no

3; n

o0;

yes

Fin

d e

ach

cro

ss p

rod

uct

. Th

en v

erif

y th

at t

he

resu

ltin

g v

ecto

r is

per

pen

dic

ula

r to

th

e g

iven

vec

tors

.7.

�1, 3

, 4��

��1,

0,�

1�8.

�3, 1

,�6�

��

2, 4

, 3�

��3,

�3,

3�;

yes

�27,

3, 1

4�; y

es

9.�3

, 1, 2

��2

,�3,

1�

10.

�4,�

1, 0

��5

,�3,

�1�

�7, 1

,�11

�; ye

s�1

, 4,�

7�; y

es

11.�

�6,

1, 3

��

2,�

2, 1

�12

.�0

, 0, 6

��3

,�2,

�4�

�7, 0

, 14�

; yes

�12,

18,

0�;

yes

13.P

hys

ics

Jan

na

is u

sin

g a

forc

e of

100

pou

nds

to

push

a c

art

up

a ra

mp.

Th

e ra

mp

is 6

fee

t lo

ng

and

is a

t a

30°

angl

e w

ith

th

eh

oriz

onta

l. H

ow m

uch

wor

k is

Jan

na

doin

g in

th

e ve

rtic

al

dire

ctio

n?

(Hin

t: U

se t

he

sin

e ra

tio

and

the

form

ula

W�

F��

d�.)

300

ft-l

b

8-4





© G

lenc

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cGra

w-H

ill33

1A

dva

nced

Mat

hem

atic

al C

once

pts

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Enr

ichm

ent

8-5

Lin

ea

rly

De

pe

nd

en

t Ve

cto

rsT

he

zero

vec

tor

is �0

, 0�i

n t

wo

dim

ensi

ons,

an

d �0

, 0, 0

�in

th

ree

dim

ensi

ons.

Ase

t of

vec

tors

is c

alle

d li

nea

rly

dep

end

ent

if a

nd

only

if t

her

eex

ist

scal

ars,

not

all

zer

o, s

uch

th

at a

lin

ear

com

bin

atio

n o

f th

e ve

ctor

s yi

elds

a z

ero

vect

or.

Exa

mp

leA

re t

he

vect

ors

�–1,

2, 1

�, �1

, –1,

2�,

and

�0, –

2, –

6�li

nea

rly

depe

nde

nt?

Sol

ve a

�–1,

2, 1

��b�

1,– 1

, 2��

c�0,

– 2,–

6��

�0, 0

, 0�.

– a�

b�

02a

�b

�2c

�0

a�

2b�

6c �

0

Th

e ab

ove

syst

em d

oes

not

hav

e a

un

iqu

e so

luti

on. A

ny

solu

tion

mu

st s

atis

fy t

he

con

diti

ons

that

a�

b�

2c.

Hen

ce, o

ne

solu

tion

is a

�1,

b�

1, a

nd

c�

.

�–1,

2, 1

��1

,–1,

2��

�0,–

2,– 6

��0

, 0, 0

�, so

th

eth

ree

vect

ors

are

lin

earl

y de

pen

den

t.

Det

erm

ine

wh

eth

er t

he

giv

en v

ecto

rs a

re li

nea

rly

dep

end

ent.

Wri

te y

es o

r n

o. If

th

ean

swer

is y

es, g

ive

a lin

ear

com

bin

atio

n t

hat

yie

lds

a ze

ro v

ecto

r.

1.�–

2, 6

�, �1

,–3�

2.�3

, 6�,

�2, 4

�ye

s; �–

2, 6

��2�

1, –

3��

�0, 0

�ye

s; 2

�3,6

��3�

2,4�

��0

, 0�

3.�1

, 1, 1

�, �–

1, 0

, 1�,

�1,–

1,– 1

�4.

�1, 1

, 1�,

�–1,

0, 1

�, �–

3,– 2

,–1�

noye

s; 2

�1, 1

, 1��

�–1,

0, 1

��–

3, –

2, –

1��

�0, 0

, 0�

5.�2

,–4,

6�,

�3,–

1, 2

�, �–

6, 8

, 10�

6.�1

,–2,

0�,

�2, 0

, 3�, �– 1

, 1,

�no

no9 � 4

1 � 2

© G

lenc

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w-H

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0A

dva

nced

Mat

hem

atic

al C

once

pts

Ap

plic

atio

ns

with

Ve

cto

rs

Mak

e a

sket

ch t

o sh

ow t

he

giv

en v

ecto

rs.

1.a

forc

e of

97

new

ton

s ac

tin

g on

an

obj

ect

wh

ile

a fo

rce

of 3

8 n

ewto

ns

acts

on

th

e sa

me

obje

ct a

t an

an

gle

of 7

0°w

ith

th

e fi

rst

forc

e

2.a

forc

e of

85

pou

nds

du

e n

orth

an

d a

forc

e of

100

pou

nds

du

e w

est

acti

ng

on t

he

sam

e ob

ject

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

the

resu

ltan

t ve

ctor

for

eac

hd

iag

ram

.3.

4.

281.

78 N

; 27.

47°

11.3

9 N

; 50.

74°

5.W

hat

wou

ld b

e th

e fo

rce

requ

ired

to

push

a 2

00-p

oun

d ob

ject

up

ara

mp

incl

ined

at

30°

wit

h t

he

grou

nd?

at le

ast

100

lb

6.N

adia

is p

ull

ing

a ta

rp a

lon

g le

vel g

rou

nd

wit

h a

for

ce o

f 25

pou

nds

dir

ecte

d al

ong

the

tarp

. If

the

tarp

mak

es a

n a

ngl

e of

50°

wit

h t

he

grou

nd,

fin

d th

e h

oriz

onta

l an

d ve

rtic

al c

ompo

nen

ts o

fth

e fo

rce.

16.0

7 lb

; 19.

15 lb

7.A

via

tion

Api

lot

flie

s a

plan

e ea

st f

or 2

00 k

ilom

eter

s, t

hen

60°

sou

th o

f ea

st f

or 8

0 ki

lom

eter

s. F

ind

the

plan

e's

dist

ance

an

ddi

rect

ion

fro

m t

he

star

tin

g po

int.

249.

80 k

m; 1

6.10

°so

uth

of

east

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

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____

____

____

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____

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8-5



© G

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dva

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once

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____

____

____

____

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8-6

Usi

ng

Pa

ram

etr

ic E

qu

atio

ns

to F

ind

th

eD

ista

nc

e f

rom

a P

oin

t to

a P

lan

eYo

u c

an u

se p

aram

etri

c eq

uat

ion

s to

hel

p yo

u f

ind

the

dist

ance

fro

m a

poin

t n

ot o

n a

pla

ne

to a

giv

en p

lan

e.

Exa

mp

le

Fin

d t

he

dis

tan

ce f

rom

P(�

1, 1

, 0)

to t

he

pla

ne

x�

2y�

z�

4.

Use

th

e co

effi

cien

ts o

f th

e eq

uat

ion

of

the

plan

e an

d th

eco

ordi

nat

es o

f th

e po

int

to w

rite

th

e ra

tios

bel

ow.

��

Th

e de

nom

inat

ors

of t

hes

e ra

tios

rep

rese

nt

a ve

ctor

th

atis

per

pen

dicu

lar

to t

he

plan

e, a

nd

pass

es t

hro

ugh

th

egi

ven

poi

nt.

Set

teq

ual

to

each

of

the

abov

e ra

tios

. Th

en,

t =

,

t =

, a

nd

t =

.

So,

x�

t�1,

y�

2t�

1, a

nd

z�

– tar

e pa

ram

etri

c eq

uat

ion

s of

th

e li

ne.

Su

bsti

tute

th

ese

valu

es in

to t

he

equ

atio

n o

f th

e pl

ane.

(t�

1)�

2(2t

�1)

�(–

t)�

4S

olve

for

t:

6t�

1�

4

t�

Th

is m

ean

s th

at t

�at

th

e po

int

of

inte

rsec

tion

of

the

vect

or a

nd

the

plan

e.

Th

e po

int

of in

ters

ecti

on is

��

1, 2�

��1,

��

Use

th

e di

stan

ce f

orm

ula

:

d�� 1 �� 1 2� � ��2 �� (1 �� 2 �)2 �� 0 �� 1 2� � ��2 �

1.2

un

its

Fin

d t

he

dis

tan

ce f

rom

th

e g

iven

poi

nt

to t

he

giv

en p

lan

e. R

oun

d y

our

answ

ers

to t

he

nea

rest

ten

th.

1.fr

om (

2, 0

, –1)

to

x�

2y�

z�

32.

from

(1,

1, –

1) t

o 2x

�y

�3z

�5

0.8

unit

0.3

unit

1 � 21 � 2

1 � 2

1 � 21 � 2

z�

0�

–1y

�1

�2

x +

1�

1

z�

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– 1y

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�2

x�

1�

1 or ��

, 2, �

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1 � 2

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____

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Vec

tors

an

d P

ara

me

tric

Eq

ua

tio

ns

Wri

te a

vec

tor

equ

atio

n o

f th

e lin

e th

at p

asse

s th

rou

gh

poi

nt

Pan

d is

par

alle

l to

a�. T

hen

wri

te p

aram

etri

c eq

uat

ion

s of

th

e lin

e.1.

P(�

2, 1

), a�

��3

,�4�

2.P

(3, 7

), a�

��4

, 5�

�x�

2, y

�1�

�t�

3,�

4��x

�3,

y�

7��

t�4,

5�

x�

�2

�3t

x�

3�

4t

y�

1�

4t

y�

7�

5t

3.P

(2,�

4), a�

��1

, 3�

4.P

(5,�

8), a�

��9

, 2�

�x�

2, y

�4�

�t

�1, 3

��x

�5,

y�

8��

t�9,

2�

x�

2�

tx

�5

�9t

y�

�4

�3t

y�

�8

�2t

Wri

te p

aram

etri

c eq

uat

ion

s of

th

e lin

e w

ith

th

e g

iven

eq

uat

ion

.5.

y�

3x �

86.

y�

�x

�4

x�

tx

�t

y�

3t�

8y

��

t�

4

7.3x

�2y

�6

8.5x

�4y

�20

x�

tx

�t

y�

�3 2� t�

3y

��

�5 4� t�

5

Wri

te a

n e

qu

atio

n in

slo

pe-

inte

rcep

t fo

rm o

f th

e lin

e w

ith

th

eg

iven

par

amet

ric

equ

atio

ns.

9.x

�2t

�3

10.

x�

t�5

y�

t�4

y�

�3t

y�

�1 2� x�

�1 21 �y

��

3x�

15

11.P

hys

ica

l E

du

cati

onB

rett

and

Cha

d ar

e pl

ayin

g to

uch

foot

ball

in g

ym c

lass

. Bre

tt h

as t

o ta

g C

had

befo

re h

e re

ache

s a

20-y

ard

mar

ker.

Cha

d fo

llow

s a

path

def

ined

by

�x �

1, y

�19

��t�

0, 1

�,an

d B

rett

foll

ows

a pa

th d

efin

ed b

y�x

�12

, y�

0��

t��

11, 1

9�.

Wri

te p

aram

etri

c eq

uati

ons

for

the

path

s of

Bre

tt a

nd C

had.

Wil

lB

rett

tag

Cha

d be

fore

he

reac

hes

the

20-y

ard

mar

ker?

Cha

d x

�1,

y�

19�

t; B

rett

x�

12�

11t,

y�

19t;

yes

8-6



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____

____

____

____

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____

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____

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____

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8-7

Co

ord

ina

te E

qu

atio

ns

of

Pro

jec

tile

sT

he

path

of

a pr

ojec

tile

aft

er it

is la

un

ched

is a

par

abol

a w

hen

gra

phed

on a

coo

rdin

ate

plan

e.

Th

e pa

th a

ssu

mes

th

at g

ravi

ty is

th

e on

ly f

orce

act

ing

on t

he

proj

ecti

le.

Th

e eq

uat

ion

of

the

path

of

a pr

ojec

tile

on

th

e co

ordi

nat

e pl

ane

is

give

n b

y,

y�

– ��x2

�(t

an �

)x,

wh

ere

gis

th

e ac

cele

rati

on d

ue

to g

ravi

ty, 9

.8 m

/s2

or 3

2 ft

/s2 ,

v 0

is t

he

init

ial v

eloc

ity,

an

d �

is t

he

angl

e at

wh

ich

th

e pr

ojec

tile

is f

ired

.

Exa

mp

le

Wri

te t

he

equ

atio

n o

f a

pro

ject

ile

fire

d a

t an

an

gle

of 1

0°to

th

e h

oriz

onta

l w

ith

an

in

itia

l ve

loci

ty o

f12

0 m

/s.

y�

– ��x2

�(t

an 1

0°)x

y�

–0.

0003

5x2

�0.

18x

Fin

d t

he

eq

uat

ion

of

the

pat

h o

f ea

ch p

roje

ctile

.

1.a

proj

ecti

le f

ired

at

80°

to t

he

2.a

proj

ecti

le f

ired

at

40°

to t

he

hor

izon

tal w

ith

an

init

ial v

eloc

ity

hor

izon

tal w

ith

an

init

ial v

eloc

ity

of 2

00 f

t/s

of 1

50 m

/sy

�–

0.01

3x2

�5.

67x

y�

–0.

0003

7x2

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84x

9.8

��

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120)

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°

g�

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Mo

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ara

me

tric

Eq

ua

tio

ns

Fin

d t

he

init

ial h

oriz

onta

l an

d v

erti

cal v

eloc

ity

for

each

sit

uat

ion

.1.

a so

ccer

bal

l kic

ked

wit

h a

n in

itia

l vel

ocit

y of

39

feet

per

sec

ond

atan

an

gle

of 4

4°w

ith

th

e gr

oun

d28

.05

ft/s

, 27.

09 f

t/s

2.a

toy

rock

et la

un

ched

fro

m le

vel g

rou

nd

wit

h a

n in

itia

l vel

ocit

y of

63 f

eet

per

seco

nd

at a

n a

ngl

e of

84°

wit

h t

he

hor

izon

tal

6.59

ft/

s, 6

2.65

ft/

s

3.a

foot

ball

th

row

n a

t a

velo

city

of

10 y

ards

per

sec

ond

at a

n a

ngl

eof

58°

wit

h t

he

grou

nd

5.30

yd

/s, 8

.48

yd/s

4.a

golf

bal

l hit

wit

h a

n in

itia

l vel

ocit

y of

102

fee

t pe

r se

con

d at

an

angl

e of

67°

wit

h t

he

hor

izon

tal

39.8

5 ft

/s, 9

3.89

ft/

s

5.M

odel

Roc

ket

ryM

anu

el la

un

ches

a t

oy r

ocke

t fr

om g

rou

nd

leve

l wit

h a

n in

itia

l vel

ocit

y of

80

feet

per

sec

ond

at a

n a

ngl

e of

80°

wit

h t

he

hor

izon

tal.

a.W

rite

par

amet

ric

equ

atio

ns

to r

epre

sen

t th

e pa

th o

f th

e ro

cket

.x

�80

tco

s 80

°; y

�80

tsi

n 80

°�

16t2

b.

How

lon

g w

ill i

t ta

ke t

he

rock

et t

o tr

avel

10

feet

hor

izon

tall

yfr

om it

s st

arti

ng

poin

t? W

hat

wil

l be

its

vert

ical

dis

tan

ce a

tth

at p

oin

t?0.

72 s

; 48.

43 f

t

6.S

por

tsJe

ssic

a th

row

s a

jave

lin

fro

m a

hei

ght

of 5

fee

t w

ith

an

init

ial v

eloc

ity

of 6

5 fe

et p

er s

econ

d at

an

an

gle

of 4

5°w

ith

th

egr

oun

d.a.

Wri

te p

aram

etri

c eq

uat

ion

s to

rep

rese

nt

the

path

of

the

jave

lin

.x

�65

tco

s 45

°;y

�65

tsi

n 45

°�

16t2

�5

b.

Aft

er 0

.5 s

econ

ds, h

ow f

ar h

as t

he

jave

lin

tra

vele

d h

oriz

onta

lly

and

vert

ical

ly?

22.9

8 ft

; 23.

98 f

t

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

8-7



© G

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Mat

hem

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once

pts

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ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

8-8

Sp

he

ric

al C

oo

rdin

ate

sT

her

e ar

e m

any

coor

din

ate

syst

ems

for

loca

tin

g a

poin

t in

th

etw

o-di

men

sion

al p

lan

e. Y

ou h

ave

stu

died

on

e of

th

e m

ost

com

-m

on s

yste

ms,

rec

tan

gula

r co

ordi

nat

es. T

he

mos

t co

mm

only

use

d th

ree-

dim

ensi

onal

coo

rdin

ate

syst

ems

are

the

exte

nde

dre

ctan

gula

r sy

stem

, wit

h a

n a

dded

z-a

xis,

an

d th

e sp

her

ical

coor

din

ate

syst

em, a

mod

ific

atio

n o

f po

lar

coor

din

ates

.

Not

e th

at t

he

orie

nta

tion

of

the

axes

sh

own

is

a d

iffe

ren

t pe

rspe

ctiv

e th

an t

hat

use

d i

n y

our

text

book

.

Poi

nt

P(d

, �, �

) in

th

ree-

dim

ensi

onal

spa

ce is

loca

ted

usi

ng

thre

e sp

her

ical

coo

rdin

ates

:d

�di

stan

ce f

rom

ori

gin

��

angl

e re

lati

ve t

o x

-axi

s�

�an

gle

rela

tive

to

y-ax

is

Th

e fi

gure

at

the

righ

t sh

ows

poin

t Q

wit

h r

ecta

ngu

lar

coor

-di

nat

es (

2, 5

, 6).

1.F

ind

OA

and

AB

.2;

62.

Fin

d O

Bby

usi

ng

the

Pyt

hag

orea

n t

heo

rem

.2

�1�0�

3.F

ind

QB

.5

4.F

ind

d.

�6�5�

5.U

se in

vers

e tr

igon

omet

ric

fun

ctio

ns

to f

ind

�an

d �

to t

he

nea

rest

degr

ee.

Wri

te t

he

sph

eric

al c

oord

inat

es o

f Q

.(�

6�5�, 7

2°, 5

2°)

Fin

d t

he

sph

eric

al c

oord

inat

es o

f th

e p

oin

t w

ith

th

e g

iven

rec

tan

gu

lar

coor

din

ates

.R

oun

d d

ista

nce

s to

th

e n

eare

st t

enth

an

d a

ng

les

to t

he

nea

rest

deg

ree.

6.(4

, 12,

3)

(13,

37°

, 23°

) 7.

(– 2

, –3,

–1)

(3.7

, 27 °

, 143

°)8.

(a, b

, c)

��a�2 ��

b�2 ��

c�2 �,

arc

tan��

, ar

cco

s�

��b

��

�a�

2 ��

b�2 ��

��c�

2 ��c � a

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E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Tra

nsf

orm

atio

n M

atr

ice

s in

Th

ree

-Dim

en

sio

na

l S

pa

ce

Wri

te t

he

mat

rix

for

each

fig

ure

.1.

2.

�

�

Tran

slat

e th

e fi

gu

re in

Qu

esti

on 1

usi

ng

th

e g

iven

vec

tors

. Gra

ph

eac

h im

age

and

wri

te t

he

tran

slat

ed m

atri

x.3.

a� �

1, 2

, 0�

4.b�

��1,

2,�

2�

�

�

Tran

sfor

m t

he

fig

ure

in Q

ues

tion

2 u

sin

g e

ach

mat

rix.

Gra

ph

eac

him

age

and

des

crib

e th

e re

sult

.5.

�

6.

�

dim

ensi

ons

incr

ease

d

refle

ctio

n o

ver

xy-p

lane

by

a fa

cto

r o

f 2

0 0�

1

0 1 0

1 0 0

0 0 2

0 2 0

2 0 0

1 4�

2

1 2 0

1 2�

2

�1 4 0

�1 4

�2

�1 2 0

1 4 0

�1 2

�2

3 4 0

3 2 2

3 2 0

1 4 2

1 4 0

1 2 2

3 4 2

1 2 0

0 0 0

�1 1

�1

1 1�

1

1�

1�

1

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2 2 0

2 0 2

2 0 0

0 2 2

0 2 0

0 0 2

2 2 2

0 0 0

8-8


Page 341

1. B

2. B

3. C

4. A

5. D

6. B

7. A

8. C

9. A

10. D

11. B

Page 342

12. C

13. B

14. B

15. A

16. D

17. B

18. C

19. B

20. B

Bonus: A

Page 343

1. A

2. C

3. A

4. C

5. B

6. B

7. D

8. D

9. B

10. A

11. B

Page 344

12. D

13. C

14. D

15. B

16. A

17. D

18. C

19. B

20. B

Bonus: A

Chapter 8 Answer KeyForm 1A Form 1B


Chapter 8 Answer Key

Page 345

1. B

2. C

3. A

4. C

5. C

6. B

7. D

8. A

9. C

10. A

11. C

Page 346

12. B

13. C

14. A

15. D

16. B

17. C

18. A

19. D

20. A

Bonus: B

Page 347

1. 10.57 m, 4.27 m

2. 313.6 in.

3. 4.2 cm, 43�

4. 0.1 cm, 223�

5. ��1.9, 8.9�; 9.10

18.5 N,6. 2.2� above east

7. ��23

�, ��34

��8. ��8

21�, ��19

83�, 4�

9. ��131�, �1

43�, �1

21��

��8.9, 7.8, �10.4�;10. 15.75

11. �2.9i� � 4.4 j� � 2k�

12. 0; yes

13. �8; no

14. �� 365�, 14, 14�

15. �20, �421�, �27

87��

Page 34816. 1 N, 313�

17. 30.6 lb

18. 26.6 N, 231.5�

�x � �32

�, y � 5� � t �2, 3�;x � �3

2� � 2t,

19. y � �5 � 3t

20. x � t, y � ��34

�t � 3

21. x � t, y � �6t � 15

22. y � �4x � 20

23. y � �2x � �343�

24.after 2.1 secondsA� (2, �3, �4), B� (2, 0, �4), C� (2, 0, 0), D� (2, �3, 0), E � (�1, �3, �4), F � (�1, 0, �4), G� (�1, 0, 0),

25. H� (�1, �3, 0)

x � t,Bonus: y � ��1

2�t � �

152�

Form 1C Form 2A


Chapter 8 Answer KeyForm 2B Form 2C

Page 349

1. 9.99 m, 0.35 m

2. 8.5 ft

3. 6.0 cm, 219�

4. 2.2 cm, 43�

5. ��1, 15�; 15.03

17.5 N; 70.2�6. below east

7. �7, �21�

8. �28, �16, 15�

9. �3, � �151�, 1�

10. ��7, 10, �7�; 14.07

11. �9i� � 5j� � 2k�

12. �12; no

13. 6; no

14. �18, 48, 28�

15. ��10, 24, 47�

Page 350

16. 13.7 N; 65.4�

17. 114.1 lb

18. 38.7 N, 228.8�

�x � 1, y� �t��3, �7�; x � 1 � 3t,

19. y � �7t

20. x � t, y � �t � 3

x � t,21. y � ��1

2�t � �

45�

22. y � �2x � 8

23. y � �2x � 8

24. about 6.19 ft

A�(2, 0, 2), B�(2, �2, 2),C�(2, �2, �2), D�(2, 0, �2), E�(0, 0, 2), F�(0, �2, 2),G�(0, �2, �2),

25. H�(0, 0, �2)

x � t,Bonus: y � �4t � 6

Page 351

1. 4.33 m, 2.5 m

2. �43

� cm

3. 3.9 cm, 49�

4. 3.8 cm, 71�

5. ��4, �1�; 4.12

56.5 N; 43.5�6. above east

7. ��3, 11�

8. ��1, �2, 1�

9. �1, �10, 7�

10. ��4, �3, �4�; 6.40

11. 4i� � 2 j� � 5k�

12. 0; yes

13. 0; yes

14. �5, 13, 1�15. �2, 4, 4�

Page 352

16. 5.0 N; 36.0�

17. 7.0 N, 13.2 N

18. 23.1 N, 284.5�

�x � 3, y � 2� �

t��2, 6�; x � �3 � 2t,19. y � 2 � 6t

20. x � t, y � 4t

21. x � t, y � 2t � 1

22. y � 2x

23. y � �21�x � 5

24. about 0.68 ftA�(3, �1, 2), B�(3, �1, 4), C�(3, 1, 4),D�(3, 1, 2), E �(1, �1, 2),F�(1, �1, 4), G�(1, 1, 4),

25. H�(1, 1, 2)

Bonus: x � t, y � 3t


CHAPTER 8 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts vector addition, subtraction, cross multiplication, inner product, and parametric equations.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts vector addition, with Minor subtraction, cross multiplication, dot product, andFlaws parametric equations.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Diagrams and graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts vector Satisfactory, addition, subtraction, cross multiplication, dot product,with Serious and parametric equations.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Diagrams and graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts vector addition, subtraction, cross multiplication, dot product,and parametric equations.

• May not use appropriate strategies to solve problems. • Computations are incorrect.• Written explanations are not satisfactory.• Diagrams and graphs are not accurate or appropriate.• Does not satisfy requirements of problems.




Page 353

1a.

1b. a� � b� � a� � (� b�), as shown in thefigure below.

1c. Yes. They are the same diagonal of a parallelogram.

1d. No. a� � b� and b� � a� are shown inthe figures below.

1e. Add the first terms of each vectortogether, and then add the secondterms together. These termsrepresent the horizontal and verticalcomponents of the resultant vector,respectively.c� � d� � ��3 � �8, 1 � (�11)�, or��11, �10�The magnitude of c� � d� is �(��1�1�)2� �� (��1�0�)2�, or about 14.9.

1f. Sample answer: �1, 2, 3� ��3, 3, 0� � �4, �1, 3�; �4, �1, 3� �4i� � j� � 3k�

1g. Sample answer: �3, 7�; The vectorsare perpendicular because their dotproduct is zero.a1b1 � a2b2 � 7 � 3 � (�3)7 � 0

1h. � � � 0i� � 0j� � 5k�

2a. x � �2 � 3ty � 4 � t

2b. Sample answer: b� � �6, �2�, (1, 3)

3a.t(56)sin 30� � �1

2�(32)t2 �8 � 0

4t2 �7t � 2 � 0(4t � 1)(t � 2) � 0

t � 2The ball hits the ground after 2seconds.

3b. Distance: x � (2)(56) ��23��, or

about 97 feet

4. Sample answer: The vectors a� � �1, 0�and b� � �0, �1� are perpendicularbecause their inner product isa1b1 � a2b2 � 1(0) � 0(�1) or 0; a� � �5, 5�, and b� � �5, �5� areperpendicular because their innerproduct is a1b1 � a2b2 � 5(5) �

5(�5) � 25 � 25 � 0.

k�00

j�13

i�21

Open-Ended Assessment


Mid-Chapter TestPage 354

1. 7.05 in., 9.71 in.

2. 39.6 cm

3. 6.6 cm; 64�

4. 5.6 cm; 260�

5. ��4, 2�; 4.47

6. �3i� � j�

7. about 8.3 mph

8. ��15, 20�

9. �3, 4�

10. �15, �24�

11. �4, 3, �12�; 13

12. �5i� � 2j�

13. ��8, 10, �1�

14. �9, �30, �5�

15. ��52

�, �2, �4�

16. �4; no

17. 0; yes

18. �5, �45, 21�

19. �14, 16, 10�

Sample answer:20. �3, 24, 12�

Quiz APage 355

1. 12.93 mm, 1.36 mm

2. 12.6 m

3. 5.5 cm; 29�

4. 5.9 cm; 187�

5. ��7, �5�; 8.60

6. �15i� � 5j�

7. 225.25 lb

8. ��15, 20�

9. �21, �24�

10. �6, �10�

Quiz BPage 355

1. �5, 0, �12�; 13

2. �3i� � 5j� � 4k�

3. �3, ��131�, 10�

4. 9; no

�18, 38, 25�; 5. both inner products � 0

Quiz CPage 356

1. 11.4 N; 50.7�

2. 19.15 N, 16.07 N

3. 72.7 N, 234.9�

�x � 1, y � 3� � t��2, 4�;4. x � 1 � 2t, y � �3 � 4t

5. x � t, y � 6t � 2

6. x � t, y � �25

�t � �54�

7. y � ��16

�x � �136�

8. y � ��43

�x � �334�

Quiz DPage 356

x � 65t cos 35�,1. y � 65t sin 35� � 16t2; yes

2. about 2.33 s

A�(0, 1, 0), B�(0, 3, 0), C�(0, 3, �3), D�(0, 1, �3), E�(3, 1, 0), F�(3, 3, 0),

3. G�(3, 3, �3), H�(3, 1, �3)

A�(1, �1, 1), B�(1, 1, 1), C�(1, 1, �2), D�(1, �1, �2), E�(�2, �1, 1), F�(�2, 1, 1),

4. G�(�2, 1, �2), H�(�2, �1, �2)

A�(�3, �3, 3), B�(�3, 3, 3), C�(�3, 3, �6), D�(�3, �3, �6),E�(6, �3, 3), F�(6, 3, 3),

5. G�(6, 3, �6), H�(6, �3, �6)



Page 357

1. D

2. C

3. A

4. E

5. A

6. D

7. C

8. B

9. D

10. D

Page 358

11. C

12. C

13. A

14. E

15. D

16. C

17. B

18. C

19. 12

20. 8

Page 359

1. 3

2.

A�(�2, �3), B�(2, �1),3. C�(0,4)

4.

5. ƒ(x) � �2�x� � 3

6. {x� x �2 or x 8}

7. ��32

�, �1, 4

8. {x� 0 x � �23

� or x 1}

232� � 360k�, k is an integer; 9. Sample answers: 592�, �128�

10. 12.3 square units

11. 3, 8�

12. ��6

� units to the right

13. �4�5�

�2�5

5��x � ��55��y � �5� � 0;

14. �5�; 333�

15. y � ��17

�x � �275�

Chapter 8 Answer KeySAT/ACT Practice Cumulative Review

� ��15

�

��35

�

0

�1


Unit 2 Answer KeyUnit 2 Review

1. �2�13

1�3�� 2. �51�

3. ��11�77�� 4. 1 5. 25.9

6. 5.8 7. 10.9

8. �14

� 9. �2�5

5�� 10. 1

11. no solution

12. two; B � 72� 33′, c � 14.1, and C � 42� 27′, orB � 107� 27′, c � 2.8, and C � 7� 33′

13. one; a � 40.1,c � 28.1, C � 42�

14. 14.1 15. 78.2

16. 90� 17. 135�

18. 630� 19. �105�

20. 6.3 in. 21. 0

22. 0 23. �1

24. 1 25. 2, �23��

26. none, ��5

�

27. none, 2�

28.

29.

30.

31.

32.

33. �34

� 34. �2�3

2�� 35. 2�2�

36. tan x � tan x cot2 x� sec x csc x

tan x (1 � cot2 x) � sec x csc x

��csoinsxx

��sin1

2 x��

� sec x csc x

��co1s x��sin

1x

�� sec x csc x

sec x csc x � sec x csc x

37. sin (180� � �) � tan � cos �

sin 180� cos � � cos 180� sin �� tan � cos �

0(cos �) � (�1) sin �� tan � cos �

sin � � tan � cos �

sin � ��ccoo

ss

��

�� tan � cos �

�csoins

��

� � cos � � tan � cos �

tan � cos � � tan � cos �

38. ��2� �4

�6�� 39. ��22��

40. 2 � �3� 41. �21�

42. �1275� 43. ��5� ��1

��0�2��1��44. �5 �

2�2�1�� 45. �4�

252�1��

46. 0�, 90�, 180� 47. 0�

48. 120�

49. �2�13

1�3��x � �3�13

1�3��y �

�2�13

1�3�� 0; �2�13

1�3�� ; 56�

50. �5�29

2�9��x � �2�29

2�9��y �

�8�29

2�9�� 0; �8�29

2�9��; 22�


Unit 2 Answer Key (continued)

51. �3�10

1�0��x � ��11�00��y � �7�

101�0�� 0;

�7�10

1�0��; 342�

52. 1.1 53. 3.9

54. 1.5 55. 6.0 cm, 89�

56. 2.3 cm, 1.5 cm

57. �3, �5� 58. ��1, �1�

59. �7, �13� 60. ��1, 7�

61. �1, 4, 1�; u� � i� � 4j� � k�

62. �13, �3, �7�;u� � 13i� � 3j� � 7k�

63. ��6, 9, 6�; u� � �6i� � 9j� � 6k�

64. �22, 0, �10�; u� � 22i� � 10k�

65. �14 66. 22

67. �6, 10, �2�

68. �x, y � 5�� t ��1, 5�;x � �t, y � 5 � 5t

69. �x � 4, y � 3� �t ��2, �2�; x � 4 � 2t, y � �3 � 2t

Unit 2 Test

1. true 2. 81 units2

3. �5�26

2�6��x � ��22�66��y � ��

12�36�� 0

4.

5. 15.7 in.

6. v: 11.3 cm; h: 14.4 cm

7. x � t, y � 5t � 2

8. ��6

� 9. none

10. �2 � �3� 11. 7.6

12. �2; no 13. �2�13

1�3��

14. �32�� 2�k

15.

16. ��18, 12�

17. �20i� � 3j� � 19k�

18. 40.9� 19. �2�7

7��

20. 270� 21. 231�; III

22. �tansxec

csxc x� � 1

��csoinsxx

�� sin1

x��

� 1�co

1s x�

�co

1s x�

�co

1s x�

23. �7, 22, 2�; yes

24. 96.2 cm 25. ��274�

26.

27. 140�

28. 28.2 N; 10.3 N

29. 2, ��2

�, ��2

� 30. �4�65

6�5��

31. 335.4 ft 32. ��6

�

33. a � 96.2, B � 22�, C � 32�

34. y � �13

�x � �130�

35. cos (90� � A) � �sin Acos 90� cos A � sin 90� sin A

� �sin A 0 � cos A � 1 � sin A � �sin A

�sin A � �sin A

36. y � Arccos x

37. �14

� 38. 83.8 cm2

39. 40� 40. yes

� 1

1 � 1

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BLANK

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