Chapter 8
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1 2 3 4 5 6 7 8 9 10
Advanced Mathematical Concepts
Chapter 8 Resource Masters
XXX
11 10 09 08 07 06 05 04
Contents
Vocabulary Builder . . . . . . . . . . . . . . . vii-viii
Lesson 8-7
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 335
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 337
Lesson 8-1
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 317
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Lesson 8-8
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 338
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Lesson 8-2
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 320
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Chapter 8 Assessment
Chapter 8 Test, Form 1A . . . . . . . . . . . . 341-342
Chapter 8 Test, Form 1B . . . . . . . . . . . . 343-344
Chapter 8 Test, Form 1C . . . . . . . . . . . . 345-346
Chapter 8 Test, Form 2A . . . . . . . . . . . . 347-348
Chapter 8 Test, Form 2B . . . . . . . . . . . . 349-350
Chapter 8 Test, Form 2C . . . . . . . . . . . . 351-352
Chapter 8 Extended Response
Assessment . . . . . . . . . . . . . . . . . . . . . . . 353
Chapter 8 Mid-Chapter Test . . . . . . . . . . . . . 354
Chapter 8 Quizzes A & B . . . . . . . . . . . . . . . 355
Chapter 8 Quizzes C & D. . . . . . . . . . . . . . . 356
Chapter 8 SAT and ACT Practice . . . . . 357-358
Chapter 8 Cumulative Review . . . . . . . . . . . 359
Unit 2 Review . . . . . . . . . . . . . . . . . . . . 361-362
Unit 2 Test . . . . . . . . . . . . . . . . . . . . . . . 363-366
Lesson 8-3
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 323
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Lesson 8-4
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 326
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Lesson 8-5
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 329
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Lesson 8-6
Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 332
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 334
© Glencoe/McGraw-Hill
SAT and ACT Practice Answer Sheet,
10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
SAT and ACT Practice Answer Sheet,
20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A19
iii
Advanced Mathematical Concepts
A Teacher’s Guide to Using the
Chapter 8 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the
resources you use most often. The Chapter 8 Resource Masters include the core
materials needed for Chapter 8. These materials include worksheets, extensions,
and assessment options. The answers for these pages appear at the back of this
booklet.
All of the materials found in this booklet are included for viewing and printing in
the Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-viii include a
Practice There is one master for each lesson.
student study tool that presents the key
vocabulary terms from the chapter. Students are
to record definitions and/or examples for each
term. You may suggest that students highlight or
star the terms with which they are not familiar.
These problems more closely follow the
structure of the Practice section of the Student
Edition exercises. These exercises are of
average difficulty.
When to Use These provide additional
practice options or may be used as homework
for second day teaching of the lesson.
When to Use Give these pages to students
before beginning Lesson 8-1. Remind them to
add definitions and examples as they complete
each lesson.
Enrichment There is one master for each
lesson. These activities may extend the concepts
in the lesson, offer a historical or multicultural
look 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 use
with all levels of students.
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 also
be used in conjunction with the Student Edition
as an instructional tool for those students who
have been absent.
© Glencoe/McGraw-Hill
When to Use These may be used as extra
credit, short-term projects, or as activities for
days when class periods are shortened.
iv
Advanced Mathematical Concepts
Assessment Options
Intermediate Assessment
The assessment section of the Chapter 8
Resources Masters offers a wide range of
assessment tools for intermediate and final
assessment. The following lists describe each
assessment master and its intended use.
Chapter Tests
•
Forms 1A, 1B, and 1C Form 1 tests contain
multiple-choice questions. Form 1A is
intended for use with honors-level students,
Form 1B is intended for use with averagelevel students, and Form 1C is intended for
use with basic-level students. These tests
are similar in format to offer comparable
testing situations.
Forms 2A, 2B, and 2C Form 2 tests are
composed of free-response questions. Form
2A is intended for use with honors-level
students, Form 2B is intended for use with
average-level students, and Form 2C is
intended for use with basic-level students.
These tests are similar in format to offer
comparable testing situations.
The Extended Response Assessment
includes performance assessment tasks that
are suitable for all students. A scoring
rubric is included for evaluation guidelines.
Sample answers are provided for
assessment.
© Glencoe/McGraw-Hill
•
Four free-response quizzes are included to
offer assessment at appropriate intervals in
the chapter.
•
The SAT and ACT Practice offers
continuing review of concepts in various
formats, which may appear on standardized
tests that they may encounter. This practice
includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided
on the master.
•
The Cumulative Review provides students
an opportunity to reinforce and retain skills
as they proceed through their study of
advanced mathematics. It can also be used
as a test. The master includes free-response
questions.
Answers
All of the above tests include a challenging
Bonus question.
•
A Mid-Chapter Test provides an option to
assess the first half of the chapter. It is
composed of free-response questions.
Continuing Assessment
Chapter Assessments
•
•
v
•
Page A1 is an answer sheet for the SAT and
ACT Practice questions that appear in the
Student Edition on page 549. Page A2 is an
answer sheet for the SAT and ACT Practice
master. These improve students’ familiarity
with the answer formats they may
encounter in test taking.
•
The answers for the lesson-by-lesson
masters are provided as reduced pages with
answers appearing in red.
•
Full-size answer keys are provided for the
assessment options in this booklet.
Advanced Mathematical Concepts
Chapter 8 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of every
student in a variety of ways. These worksheets, many of which are found
in the FAST FILE Chapter Resource Masters, are shown in the chart
below.
•
Study Guide masters provide worked-out examples as well as practice
problems.
•
Each chapter’s Vocabulary Builder master provides students the
opportunity to write out key concepts and definitions in their own
words.
•
Practice masters provide average-level problems for students who
are moving at a regular pace.
•
Enrichment masters offer students the opportunity to extend their
learning.
Five Different Options to Meet the Needs of
Every Student in a Variety of Ways
primarily skills
primarily concepts
primarily applications
BASIC
AVERAGE
1
Study Guide
2
Vocabulary Builder
3
Parent and Student Study Guide (online)
© Glencoe/McGraw-Hill
4
Practice
5
Enrichment
vi
ADVANCED
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Reading to Learn Mathematics
Vocabulary Builder
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
Found
on 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)
© Glencoe/McGraw-Hill
vii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Reading to Learn Mathematics
Vocabulary Builder (continued)
Vocabulary Term
Found
on Page
Definition/Description/Example
parametric equation
polyhedron
resultant
scalar
scalar quantity
standard position
unit vector
vector
vector equation
zero vector
© Glencoe/McGraw-Hill
viii
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-1
Study Guide
Geometric Vectors
The magnitude of a vector is the length of a directed line
segment. The direction of the vector is the directed angle
between the positive x-axis and the vector. When adding or
subtracting vectors, use either the parallelogram or the
triangle method to find the resultant.
Example 1
Use the parallelogram method to find the
.
sum of v and w
Copy v and w , placing the initial points
together.
as two
Form a parallelogram that has v and w
of its sides.
Draw dashed lines to represent the other two
sides.
The resultant is the vector from the vertex of v
to the opposite vertex of the
and w
parallelogram.
Use a ruler and protractor to measure the
magnitude and direction of the resultant.
The magnitude is 6 centimeters, and the
direction is 40°.
Example 2
.
3w
Use the triangle method to find 2v
3w
2v
(3w
)
2v
Draw a vector that is twice the magnitude of v
. Then draw a vector with the
to represent 2v
and three times its
opposite direction to w
. Place the initial
magnitude to represent 3w
on the terminal point of 2v
.
point of 3w
Tip-to-tail method.
Draw the resultant from the initial point of the
first vector to the terminal point of the second
3w
.
vector. The resultant is 2v
© Glencoe/McGraw-Hill
317
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-1
Practice
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.
y
5. x z
4. x 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 hour
due north when it encounters a wind blowing out of the west at
50 miles per hour. What is the magnitude of the airplane's resultant
velocity?
© Glencoe/McGraw-Hill
318
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-1
Enrichment
More Than Two Forces Acting on an Object
Three or more forces may work on an object at one time. Each of these
forces can be represented by a vector. To find the resultant vector that
acts upon the object, you can add the individual vectors two at a time.
Example
A force of 80 N acts on an object at an
angle of 70° at the same time that a
force of 100 N acts at an angle of 150 °.
A third force of 120 N acts at an angle
of 180°. Find the magnitude and
direction of the resultant force acting on
the object.
Add two vectors at a time. The order in which the vectors
are added does not matter.
Add the 80-N vector and
the 100-N vector first.
Now add the resulting vector
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
an object. A second force acts with
100 N at 110°. A third force acts with
10 N at 150°. Find the magnitude
and amplitude of the resultant
force.
© Glencoe/McGraw-Hill
2. One force acts with 75 N at 45°. A
second force acts with 90 N at 90°.
A third force acts with 120 N at 170°.
Find the magnitude and amplitude
of the resultant force.
319
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-2
Study Guide
Algebraic Vectors
Vectors can be represented algebraically using ordered pairs
of real numbers.
Example 1
Write the ordered pair that represents the vector
from X(2, 3) to Y(4, 2). Then find the
magnitude 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
y)2
(x
( y
2
1
2
1
22
(4 2) [2 (3)]
(
6
)2
52
6
1
XY is represented by the ordered pair 6, 5
1
units.
and has a magnitude of 6
Example 2
Let s 4, 2 and t 1, 3. Find each of the
following.
b. s t
a. s t
s t 4, 2 1, 3
s t 4, 2 1, 3
4 (1), 2 3
4 (1), 2 3
5, 1
3, 5
c. 4s
t
d. 3s
4s 44, 2
t 34, 2 1, 3
3s
4 4, 4 2
12, 6 1, 3
16, 8
11, 9
A unit vector in the direction of the positive
x-axis is represented by i, and a unit vector in
the direction of the positive y-axis is represented
by j. Vectors represented as ordered pairs can be
written as the sum of unit vectors.
Example 3
as the sum of unit vectors for M(2, 2)
Write MP
and P(5, 4).
as an ordered pair.
First write MP
5 2, 4 2
MP
3, 2
as the sum of unit vectors.
Then write MP
3i 2j
MP
© Glencoe/McGraw-Hill
320
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-2
Practice
Algebraic Vectors
Write the ordered pair that represents AB. Then find the
magnitude 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
w 3, 5.
v 2, 1 and 5. u
w 2v
4. u 3v
3w
6. u 4v
3v
7. u 5w
Find the magnitude of each vector, and write each vector as the
sum of unit vectors.
8. 2, 6
9. 4, 5
10. Gardening Nancy and Harry are lifting a stone statue and
moving it to a new location in their garden. Nancy is pushing the
statue with a force of 120 newtons (N) at a 60° angle with the
horizontal 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?
© Glencoe/McGraw-Hill
321
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-2
Enrichment
Basis Vectors
The expression v r
u s
w, the sum of two vectors each multiplied by
and w
.
scalars, is called a linear combination of the vectors u
Every vector v v2 can be written as a
linear combination of any two nonparallel vectors
u
and w
. The vectors u
and w
are said to form a
basis for the vector space v2 which contains
all vectors having 1 column and 2 rows.
Linear Combination
Theorem in v2
as a linear combination of
and w
.
the vectors u
Write the vector v
Example
2
5
2
3
1
4
r s –2
5
2
3
1
–4
2r s
3r 4s
–2 2r s
5 3r 4s
Solving the system of equations yields the solution
3
16
3
–
u
r –
and s – . So, v
11
11
11
16
w
.
11
Write each vector as a linear combination of the vectors u and w.
1. v
3. v
1
5
_1_
2
–1
,u
,u
–3
4
, w
0
4
© Glencoe/McGraw-Hill
, w
2
–2
2. v
4. v
, u , w _1_
2
1
322
1
–1
2
–7
,u
2
3
–1
–3
,w
1
__
4
1
4
2
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-3
Study Guide
Vectors in Three-Dimensional Space
Ordered triples, like ordered pairs, can be used to represent
vectors. Operations on vectors respresented by ordered triples
are similar to those on vectors represented by ordered pairs.
For example, an extension of the formula for the distance
between two points in a plane allows us to find the distance
between 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 to
represent the three planes.
The planes intersect at (1, 3, 1).
Example 2
Write the ordered triple that represents the
vector from X(4, 5, 6) to Y(2, 6, 3). Then find
.
the magnitude of XY
XY (2, 6, 3) (4, 5, 6)
2 (4), 6 5, 3 6
2, 1, 3
(x
2
x1)2
( y2
y
(z2
XY
)2
z
)2
1
1
[2 (4)]
(6
5) (3 6)
2
2
2
(2) (1) (3)
1
4
or 3.7
2
2
2
Example 3
if
Find an ordered triple that represents 2s 3t
s 5, 1, 2 and t 4, 3, 2.
25, 1, 2 34, 3, 2
3t
2s
10, 2, 4 12, 9, 6
22, 7, 2
Example 4
as the sum of unit vectors for A(5, 2, 3)
Write AB
and B(4, 2, 1).
First express AB as an ordered triple. Then write
.
the 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
© Glencoe/McGraw-Hill
323
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-3
Practice
Vectors in Three-Dimensional Space
Locate point B in space. Then find the magnitude of a vector from
the origin to B.
1. B(4, 7, 6)
2. B(4, 2, 6)
Write the ordered triple that represents AB . Then find the
magnitude 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
6, 8, 9.
v 2, 4, 5 and w
7. u v w
3w
9. u 4v
8. u v w
2w
10. u 5v
11. Physics Suppose that the force acting on an object can be
expressed by the vector 85, 35, 110, where each measure in
the ordered triple represents the force in pounds. What is the
magnitude of this force?
© Glencoe/McGraw-Hill
324
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-3
Enrichment
Basis Vectors in Three-Dimensional Space
The expression v
r
u s
w t
z, the sum of three vectors each
, w
,
multiplied by scalars, is called a linear combination of the vectors u
.
and z
v3 can be written as a linear combination of any
Every vector v
three nonparallel vectors. The three nonparallel vectors, which must
be linearly independent, are said to form a basis for v3, which contains
all vectors having 1 column and 3 rows.
Example
Write the vector v
1
3
1
the vectors u
–1
–4
3
r
1
3
1
s
1
–2
1
1
4
3
as a linear combination of
1
2
1
, w
t
–1
–1
1
, and z
1
1
1
.
rst
3r 2s t
rst
–1 r s t
– 4 3r 2s t
3rst
Solving the system of equations yields the solution
r 0, s 1, and t 2. So, v
w
2
z.
, w
, and z.
Write each vector as a linear combination of the vectors u
1. v
, and z
2. v
, and z
3. v
–6
–2
2
5
–2
0
1
–1
2
1
1
0
, u
,u
1
–2
3
, u
© Glencoe/McGraw-Hill
1
2
–1
, w
, w
, w
1
0
1
–1
0
1
2
2
1
0
1
1
, and z
4
2
–1
1
0
1
325
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-4
Study Guide
Perpendicular Vectors
Two vectors are perpendicular if and only if their inner
product is zero.
Example 1
5, 1, v
3, 15,
Find each inner product if u
2, 1. Is either pair of vectors perpendicular?
and w
w
b. v
w
3(2) 15(1)
v
6 (15)
21
v
a. u
v
5(3) 1(15)
u
15 15
0
and w
are not perpendicular.
v
and v
are perpendicular.
u
Example 2
3, 1, 0
Find the inner product of r and s if r
r and s perpendicular?
and s 2, 6, 4. Are r s (3)(2) (1)(6) (0)(4)
6 (6) 0
0
r and s are perpendicular since their inner
product is zero.
Unlike the inner product, the cross product of two vectors is
a vector. This vector does not lie in the plane of the given
vectors but is perpendicular to the plane containing the two
vectors.
Example 3
if v
0, 4, 1
Find the cross product of v and w
0, 1, 3. Verify that the resulting vector
and w
.
is perpendicular to v and w
i j k
w
0 4 1
v
0 1 3
41 13i 00 13j 00 41k
Expand by minors.
11i 0j 0k
11i or 11, 0, 0
Find the inner products.
11, 0, 0 0, 4, 1
11(0) 0(4) 0(1) 0
11, 0, 0 0, 1, 3)
11(0) 0(1) 0(3) 0
Since the inner products are zero, the cross
w
is perpendicular to both v
and w
.
product v
© Glencoe/McGraw-Hill
326
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-4
Practice
Perpendicular Vectors
Find each inner product and state whether the vectors are
perpendicular. 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 is
perpendicular 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
11. 6, 1, 3 2, 2, 1
10. 4, 1, 0 5, 3, 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 the
horizontal. How much work is Janna doing in the vertical
d.)
direction? (Hint: Use the sine ratio and the formula W F
© Glencoe/McGraw-Hill
327
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-4
Enrichment
Vector Equations
and c
Let a
, b,
be fixed vectors. The equation f (x) a
2x b x 2 c
defines a vector function of x. For the values of x shown, the
assigned vectors are given below.
–2
–1
0
1
c
a
2b
a
c
a
2b
4c
a
4b
x
f (x)
4c
a
4b
2
1, 1, and c
0, 1, b
2, –2, the resulting vectors for the
If a
values of x are as follows.
x
f (x)
–2
–1
12,–3 4, 1
0
1
2
0, 1 0,–3
4,–11
For each of the following, complete the table of resulting vectors.
3x
2x2 b
c
1. f (x) x 3a
x
a
1, 1 b 2, 3 c
3, –1
–1
0
1
2
5
2. f (x) 2x 2a
3x b
c
a
0, 1, 1 b 1, 0, 1
3. f (x) x 2 c
3xa
4b
3, 2, 1
a
1, 1, 1 b
x
c
1, 1, 0
3
4. f (x) x3a
xb
c
a
0,1, –2 b 1, –2, 0 c
–2, 0, 1
© Glencoe/McGraw-Hill
328
f (x)
–2
–1
0
1
x
c
0, 1, 2
f (x)
f (x)
0
1
2
3
x
f (x)
–1
0
1
3
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-5
Study Guide
Applications with Vectors
Vectors can be used to represent any quantity that has
direction and magnitude, such as force, velocity, and weight.
Example
Suppose Jamal and Mike pull on the ends of a
rope 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 and
Mike pull on the rope is 60°.
a. Draw a labeled diagram that represents
the forces.
and F
represent the two forces.
Let F
1
2
b. Determine the magnitude of the
resultant force.
First find the horizontal (x) and vertical ( y)
components of each force.
on the x-axis, the unit
Given that we place F
1
vector is 1i 0j.
are
Therefore, the x- and y-components of F
1
60i 0j.
xi yj
F
2
y
x
sin 60° cos 60° 50
50
y 50 sin 60°
x 50 cos 60°
43.3
25
25i 43.3j.
Thus, F
2
Then add the unit components.
(60i 0j) (25i 43.3j) 85i 43.3j
F
8
5
2
3
4.3
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.
43
.3
Use the tangent ratio.
tan 85
43
.3
tan1 85
27°
The direction of the resultant force is 27° with
respect to the vector on the x-axis.
© Glencoe/McGraw-Hill
329
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-5
Practice
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 west
acting on the same object
Find the magnitude and direction of the resultant vector for each
diagram.
3.
4.
5. What would be the force required to push a 200-pound object up a
ramp inclined at 30° with the ground?
6. Nadia is pulling a tarp along level ground with a force of 25
pounds directed along the tarp. If the tarp makes an angle of 50°
with the ground, find the horizontal and vertical components of
the 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 and
direction from the starting point.
© Glencoe/McGraw-Hill
330
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-5
Enrichment
Linearly Dependent Vectors
The zero vector is 0, 0 in two dimensions, and 0, 0, 0 in three
dimensions.
A set of vectors is called linearly dependent if and only if there
exist 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 b1, –1, 2 c0, –2, –6 0, 0, 0.
–a b 0
2a b 2c 0
a 2b 6c 0
The above system does not have a unique solution. Any
solution must satisfy the conditions that a b 2c.
Hence, one solution is a 1, b 1, and c 1
.
2
–1, 2, 1 1, –1, 2 0, –2, –6 0, 0, 0, so the
three vectors are linearly dependent.
Determine whether the given vectors are linearly dependent. Write yes or no. If the
answer 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,
© Glencoe/McGraw-Hill
331
9
4
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-6
Study Guide
Vectors and Parametric Equations
Vector equations and parametric equations allow us to
model movement.
Example 1
Write a vector equation describing a line passing
a 6, 1. Then
through P1(8, 4) and parallel to write parametric equations of the line.
.
Let the line through P1(8, 4) be parallel to a
For any point P2(x, y) on , P1 P2x 8, y 4.
, P1 P2 ta
,
Since P1 P2 is on and is parallel to a
for some value t. By substitution, we have
x 8, y 4 t6, 1.
Therefore, the equation x 8, y 4 t6, 1
is a vector equation describing all of the points (x, y)
through P1(8, 4).
on parallel to a
Use the general form of the parametric
equations of a line with a1, a2 6, 1
and x1, y1 8, 4.
x x1 ta1
x 8 t(6)
x 8 6t
y y1 ta2
y 4 t(1)
y4t
Parametric equations for the line are x 8 6t
and y 4 t.
Example 2
Write an equation in slope-intercept form of the
line whose parametric equations are x 3 4t
and y 3 4t.
Solve each parametric equation for t.
x 3 4t
x 3 4t
x 3 t
4
y 3 4t
y 3 4t
y3
t
4
Use substitution to write an equation for the line
without the variable t.
x 3
4
y3
4
(x 3)(4) 4( y 3)
4x 12 4y 12
yx6
© Glencoe/McGraw-Hill
Substitute.
Cross multiply.
Simplify.
Solve for y.
332
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-6
Practice
Vectors and Parametric Equations
Write a vector equation of the line that passes through point P
and is parallel to a. Then write parametric equations of the line.
2. P(3, 7), a 4, 5
1. P(2, 1), a 3, 4
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 the
given parametric equations.
9. x 2t 3
10. x t 5
yt4
y 3t
11. Physical Education Brett and Chad are playing touch football
in gym class. Brett has to tag Chad before he reaches a 20-yard
marker. Chad follows a path defined by x 1, y 19 t0, 1,
and Brett follows a path defined by x 12, y 0 t11, 19.
Write parametric equations for the paths of Brett and Chad. Will
Brett tag Chad before he reaches the 20-yard marker?
© Glencoe/McGraw-Hill
333
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-6
Enrichment
Using Parametric Equations to Find the
Distance from a Point to a Plane
You can use parametric equations to help you find the distance from a
point 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 the
coordinates of the point to write the ratios below.
x1
1
y1
2
z0
–1
The denominators of these ratios represent a vector that
is perpendicular to the plane, and passes through the
given point.
Set t equal to each of the above ratios. Then, t =
t=
y1
,
2
and t =
z0
.
–1
x+1
,
1
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) 4
Solve for t: 6t 1 4
t
1
2
1
This means that t at the point of
2
intersection of the vector and the plane.
The point of intersection is
1
1
1
2
1, 1, 2
1
2
1
2
, 2, .
or 2
2
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 the
nearest tenth.
1. from (2, 0, –1) to x 2y z 3
© Glencoe/McGraw-Hill
2. from (1, 1, –1) to 2x y 3z 5
334
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-7
Study Guide
Modeling Motion Using Parametric Equations
We can use the horizontal and vertical components of a
projectile to find parametric equations that represent the
path of the projectile.
Example 1
Find the initial horizontal and vertical velocities
of a soccer ball kicked with an initial velocity of
33 feet per second at an angle of 29° with the
ground.
x v
 cos v
x 33 cos 29°
v
x 29
v
y v
 sin v
y 33 sin 29°
v
y 16
v
The initial horizontal velocity is about 29 feet
per second and the initial vertical velocity is
about 16 feet per second.
The path of a projectile launched from the ground may be
 cos for
described by the parametric equations x tv
1
2
 sin gt for vertical
horizontal distance and y tv
2
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 of
parametric equations defining the postition of
the rock for any time t in seconds.
 cos x tv
 sin 1 gt2
y tv
2
x t(50) cos 8°
 50 m/s
y t(50) sin 8° 12(9.8)t2 v
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°
39.61
y 50(0.8) sin 8° 4.9(0.8)2
2.43
After 0.8 second, the rock has traveled about
39.61 meters horizontally and is about 2.43 meters
above the ground.
© Glencoe/McGraw-Hill
335
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-7
Practice
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 of
63 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 angle
of 58° with the ground
4. a golf ball hit with an initial velocity of 102 feet per second at an
angle of 67° with the horizontal
5. Model Rocketry Manuel launches a toy rocket from ground
level with an initial velocity of 80 feet per second at an angle of
80° 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 horizontally
from its starting point? What will be its vertical distance at
that point?
6. Sports Jessica throws a javelin from a height of 5 feet with an
initial velocity of 65 feet per second at an angle of 45° with the
ground.
a. Write parametric equations to represent the path of the
javelin.
b. After 0.5 seconds, how far has the javelin traveled horizontally
and vertically?
© Glencoe/McGraw-Hill
336
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-7
Enrichment
Coordinate Equations of Projectiles
The path of a projectile after it is launched is a parabola when graphed
on 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–
x
g
2
2v0 cos 2
2
(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 angle
of 10° to the horizontal with an initial velocity of
120 m/s.
y–
9.8
2(120)2 cos 2 10°
x
2
(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
horizontal with an initial velocity
of 200 ft/s
© Glencoe/McGraw-Hill
2. a projectile fired at 40° to the
horizontal with an initial velocity
of 150 m/s
337
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-8
Study Guide
Transformation Matrices in Three-Dimensional Space
Example 1
Find the coordinates of the vertices of the
pyramid 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)
A B C D
x 2 2 2 2
The vertex matrix for the pyramid is y 2 2 2 2
z 2 2 2 2
Example 2
E
0
0.
2
Let M represent the vertex matrix of the pyramid
in Example 1.
1 0 0
a. Find TM if T 0 1 0 .
0 0 1
b. Graph the resulting image and describe the
transformation represented by matrix T.
a. First find TM.
1 0
TM 0 1
0 0
0 2 2 2 2
0 2 2 2 2
1 2 2 2 2
A B C D 2 2 2 2
0
2 2 2 2
0
2 2 2 2
2
E
0
0
2
b. Then graph the points
represented by the resulting
matrix.
The transformation matrix reflects the image
of the pyramid over the xz-plane.
© Glencoe/McGraw-Hill
338
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-8
Practice
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 and
write the translated matrix.
4. b 1, 2, 2
3. a 1, 2, 0
Transform the figure in Question 2 using
image and describe the result.
6.
5. 2 0 0
0 2 0
0 0 2
© Glencoe/McGraw-Hill
each matrix. Graph each
1
0
0
339
0 0
1 0
0 1
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
8-8
Enrichment
Spherical Coordinates
There are many coordinate systems for locating a point in the
two-dimensional plane. You have studied one of the most common systems, rectangular coordinates. The most commonly
used three-dimensional coordinate systems are the extended
rectangular system, with an added z-axis, and the spherical
coordinate 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 using
three 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 coordinates (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 nearest
degree. 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)
© Glencoe/McGraw-Hill
340
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1A
Write the letter for the correct answer in the blank at the right of
each problem.
has a magnitude of 89.7 feet and a direction of 12° 48.
1. The vector v
Find the magnitude of its vertical component.
A. 887.47 ft
B. 19.87 ft
C. 19.38 ft
D. 87.58 ft
involving r, s, and t ?
2. What is an expression for x
A. 3r s t
B. 3r s t
D. 3r s t
C. 3r s t
1. ________
2. ________
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; 7.13
C. 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; 8.91
C. 1.8, 9.4, 15.4; 12.33
D. 1, 8.8, 1; 8.80
in 2v
2, 4
if w
u 43 w
5. ________
5. Find an ordered pair to represent u
3
3
, 2 .
and v
8
A. 14, 7
B. 54, 1
C. 14, 4
D. 54, 7
in x
6 z 1 y
if y
2, 18, 4
6. Find an ordered triple to represent x
4
5
and z 1, 3, 1 .
B. 72, 0, 45
2 4
2
5
4
, 0, 8
5
6
4
15
C. 9i 8290 j 943 k
C. 72, 0, 65
D. 72, 383, 45
as the sum of unit vectors for M 3, 5, 2 and N 6, 9, 3 .
7. Write MN
4 3
5
2
7
2
1
1
1
A. i 14 j k
B. i 14 j k
A.
4
15
2
7
D. 4 i 14 j 11
k
5
4, 5, 1 and
if a
4
3
and b
8. Find the inner product of a
1
3
b 2, 2, 2 , and state whether the vectors are perpendicular.
A. 5; no
B. 5; yes
C. 0; yes
D. 0; no
1
3
and w
if v
, 4, and w
6, 4, 4 .
9. Find the cross product of v
3
8
5
1
5
7
3
5
6
C. 1
, 3, 1
0
5 15
7 , 11
35
6
, A. 10
12
15
6. ________
7. ________
8. ________
9. ________
16
3, 11
35
6
, B. 10 12
15
16
3 , 11
35
6
, D. 10
12
15
10. Find the magnitude and direction of the
10. ________
resultant vector for the diagram at the right.
A. 8.2 N, 73° 35B. 20 N, 18° 37
C. 6.5 N, 79° 7 D. 8.2 N, 83° 48
11. ________
11. A force F 1 of 35 newtons pulls at an angle of 15° north of due east.
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 south
C. 42.2 N, 54.1° west of due south
D. 42.2 N, 27.4° west of due south
© Glencoe/McGraw-Hill
341
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1A (continued)
Write a vector equation of the line that passes through point P and is
. Then write parametric equations of the line.
parallel to a
6, 1
12. P(1, 3); a
A. x 1, y 3 t6, 1; x 1 6t, y 3 t
B. x 1, y 3 t6, 1; x 1 6t, y 3 t
C. x 1, y 3 t6, 1; x 1 6t, y 3 t
D. x 1, y 3 t6, 1; x 1 6t, y 3 t
2, 9
13. P(0, 5); a
A. x, y 5 t2, 9; x 2t, y 5 9t
B. x, y 5 t2, 9; x 2t, y 5 9t
C. x 2, y 9 t0.5; x 2, y 9 5t
D. x 2, y 9 t0, 5; x 2, y 9 5t
14. Which graph represents a line whose parametric equations are
x 2t 4 and y t 2?
A.
B.
C.
D.
13. ________
15. Write parametric equations of 3x 12 y 23.
15. ________
A. x t; y 6t 43
12. ________
14. ________
B. x t; y 6t 13
C. x t; y 6t 13
D. x t; y 6t 43
16. Write an equation in slope-intercept form of the line whose
parametric equations are x 12 t 23 and y t 34.
16. ________
7
B. y 2x 17
C. y 2x 17
D. y 2x A. y 2x 17
2
2
2
12
Darius serves a volleyball with an initial velocity of 34 feet per
second 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 ft
18. 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)
3
and 1 w
if v
2, 12, 3 Bonus:
Bonus Find the cross product of 4 v
2
7, 4, 6.
and w
4
5
2
3 , 27
5
7 C. , A. 2, 87, 527B. 62
425 , 287, 527 D. 425, 287 , 629 8
2 © Glencoe/McGraw-Hill
342
________
________
________
________
________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1B
Write the letter for the correct answer in the blank at the right of
each problem.
has a magnitude of 6.1 inches and a direction of 55°. Find
1. The vector v
the magnitude of its vertical component.
A. 5.00 in.
B. 10.64 in.
C. 7.45 in.
D. 3.50 in.
1. ________
involving r and s?
2. What is an expression for x
B. r 2s
A. r 2s
C. r 2s
D. r 2s
2. ________
3. Find the ordered pair that represents the vector from A(9, 2) to
B(6, 3). Then find the magnitude of AB.
A. 15, 1; 15.03
B. 3, 5; 5.83
C. 15, 1; 3.74
D. 3, 1; 3.16
3. ________
4. Find the ordered triple that represents the vector from A(3, 5, 6) to
B(6, 8, 6). Then find the magnitude of AB.
A. 3, 3, 0; 4.24
B. 9, 13, 12; 19.85
C. 3, 3, 0; 4.24
D. 9, 3, 0; 9.49
4. ________
in u
4w
2v
if w
3, 4
5. Find an ordered pair to represent u
and v 4, 0.
A. 20, 16
B. 4, 16
C. 10, 8
D. 22, 8
5. ________
in x
3z 5y
if y
2, 11, 5
6. Find an ordered triple to represent x
and z 2, 8, 6.
A. 4, 79, 7
B. 16, 31, 43
C. 2, 17, 1
D. 16, 7, 45
6. ________
as the sum of unit vectors for M(14, 8, 6) and N(7, 9, 2).
7. Write MN
B. 7i j 8k
A. 7i j 8k
C. 21i j 8k
D. 21i j 8k
7. ________
if a
7, 2, 4 8. ________
and b
4, 2, 2 and b
8. Find the inner product of a
and state whether the vectors are perpendicular.
A. 0; yes
B. 32; yes
C. 40; no
D. 32; no
if v
9, 4, 8 and w
6, 2, 4. 9. ________
9. Find the cross product of v and w
A. 54, 8, 32
B. 0, 12, 6
C. 32, 84, 42
D. 6, 12, 0
10. Find the magnitude and direction of the
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°
10. ________
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°
© Glencoe/McGraw-Hill
343
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1B (continued)
Write a vector equation of the line that passes through point P
and is parallel to a. Then write parametric equations of the line.
7, 6
12. P(2, 5); a
A. x 2, y 5 t7, 6; x 2 7t, y 5 6t
B. x 2, y 5 t7, 6; x 2 7t, y 5 6t
C. x 2, y 5 t7, 6; x 2 7t, y 5 6t
D. x 2, y 5 t7, 6; x 2 7t, y 5 6t
12. ________
1, 8
13. P(0, 3); a
A. x, y 3 t1, 8; x t, y 3 8t
B. x 1, y 8 t(0, 3); x 1, y 8 3t
C. x, y 3 t1, 8; x t, y 3 8t
D. x 1, y 8 t0, 3; x 1, y 8 3t
13. ________
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.
B. x t; y 14t 54
A. x t; y 4t 54
C. x t; y 4t 54
D. x t; y 14t 54
15. ________
16. Write an equation in slope-intercept form of the line whose
parametric equations are x 3t 8 and y 2t 9.
1 D. y 2x 1
1
A. y 23x 433 B. y 23x 433 C. y 23x 13
3
3
16. ________
Aaron kicked a soccer ball with an initial velocity of 39 feet per
second at an angle of 44° with the horizontal.
17. After 0.9 second, how far has the ball traveled horizontally?
A. 24.4 ft
B. 12.3 ft
C. 11.4 ft
D. 25.2 ft
17. ________
18. After 1.5 seconds, how far has the ball traveled vertically?
A. 6.1 ft
B. 40.6 ft
C. 4.6 ft
D. 42.1 ft
18. ________
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
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?
A. E′(1, 1, 0)
B. E′(1, 1, 0)
C. E′(1, 1, 0)
D. E′(2, 1, 0)
2w
if v
1, 5, 3 and w
7, 5, 6.
Bonus Find 3v
A. 270, 162, 180
B. 270, 90, 240
C. 270, 90, 240
D. 270, 162, 180
© Glencoe/McGraw-Hill
344
19. ________
20. ________
Bonus: ________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1C
Write the letter for the correct answer in the blank at the right of each problem.
has a magnitude of 5 inches and a direction of 32°.
1. ________
1. The vector v
Find the magnitude of its vertical component.
A. 4.24 in
B. 2.65 in
C. 2.79 in
D. 31.88 in
involving r and s ?
2. What is an expression for x
B. r s
A. r s
C. r s
D. r s
2. ________
3. Find the ordered pair that represents the vector from
A(1, 2) to B(0, 3). Then find the magnitude of AB.
A. 1, 1; 1.41
B. 1, 1; 2
C. 1, 1; 1.41
D. 1, 1; 2
3. ________
4. Find the ordered triple that represents the vector from A(4, 2, 1) to
B(3, 0, 5). Then find the magnitude of AB.
A. 7, 2, 4; 8.31
B. 1, 2, 4; 4.58
C. 1, 2, 4; 4.58
D. 7, 2, 6; 9.43
4. ________
in u
2w
v
if w
2, 4 and
5. Find an ordered pair to represent u
3, 1.
v
A. 7, 7
B. 1, 7
C. 7, 7
D. 1, 7
5. ________
in x
3y
z if y
2, 1, 5
6. Find an ordered triple to represent x
and z 1, 6, 6.
A. 7, 3, 9
B. 5, 3, 9
C. 5, 9, 9
D. 7, 3, 21
6. ________
as the sum of unit vectors for M(2, 3, 6) and N(1, 5, 2).
7. Write MN
B. i 2 j 4k
A. i 2 j 8 k
C. 3i 2 j 4 k
D. 3i 2 j 8 k
7. ________
if a
4, 2, 5 and
and b
3, 0, 1 and b
8. Find the inner product of a
state whether the vectors are perpendicular.
A. 7; no
B. 0; yes
C. 7; yes
D. 0; no
8. ________
and w
if v
1, 2, 4 and w
3, 1, 5. 9. ________
9. Find the cross product of v
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
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°
10. ________
11. A 22-newton force acting at 48° and a 65-newton
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°
11. ________
© Glencoe/McGraw-Hill
345
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 1C (continued)
Write a vector equation of the line that passes through point P and
. Then write parametric equations of the line.
is parallel to a
2, 5
12. P(1, 3); a
A. x 1, y 3 t2, 5; x 1 2t, y 3 2t
B. x 1, y 3 t2, 5; x 1 2t, y 3 5t
C. x 1, y 3 t2, 5; x 1 2t, y 3 5t
D. x 1, y 3 t2, 5; x 1 2t, y 3 2t
2, 5
13. P(1, 4); a
A. x 2, y 5 t1, 4; x 2 t, y 5 4t
B. x 2, y 5 t1, 4; x 2 t, y 5 4t
C. x 1, y 4 t2, 5; x 1 2t, y 4 5t
D. x 1, y 4 t2, 5; x 1 2t, y 4 5t
14. Which graph represents a line whose parametric equations are
x t 2 and y t 2?
A.
B.
C.
D.
15. Write parametric equations of y 2x 3.
B. x t; y 2t 3
A. x t; y 12 t 3
12. ________
13. ________
14. ________
15. ________
C. x t; y 12 t 3
D. x t; y 2t 3
16. Write an equation in slope-intercept form of the line whose
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?
A. 27.9 ft
B. 123.8 ft
C. 79.7 ft
D. 97.7 ft
18. After 3 seconds, how far has the ball traveled vertically?
A. 137.7 ft
B. 119.6 ft
C. 233.7 ft
D. 52.6 ft
16. ________
17. ________
18. ________
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)
and 2w
if v
2, 4, 1 and
Bonus: ________
Bonus Find the cross product of v
1, 2, 5.
w
A. 44, 22, 16 B. 36, 22, 16 C. 36, 22, 16 D. 36, 22, 0
© Glencoe/McGraw-Hill
346
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2A
has a magnitude of 11.4 meters and a direction 1. __________________
1. The vector v
of 248°. Find the magnitude of its vertical and horizontal
components.
has a magnitude of 89.6 inches. If v
7 u
,
2. The vector u
2
what is the magnitude of v?
2. __________________
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.
2 a
1 b
3. 3 a
2
3
3. __________________
2 b
4. 12 a
5
4. __________________
5. Write the ordered pair that represents the vector from
A(1.8, 3.8) to B(0.1, 5.1). Then find the magnitude of AB .
5. __________________
of 18.8 newtons pulls at an angle of 12° above
6. A force F
1
of 3.2 newtons pulls at an angle of
due east. A force F
2
88° below due east. Find the magnitude and direction of
the resultant force.
6. __________________
Find an ordered pair or ordered triple to represent u in
1
3
1
2, , r 1, , 2,
each equation if v 0, , w
4
4
and s 10, 6,
.
3
4
7. __________________
2
8. __________________
v
1 w
8. u
1 r 4s
7. u
3
2
2 s 3r
9. u
3
9. __________________
10. Write the ordered triple that represents the vector from
A(5.1, 0.8, 9) to B(3.8, 7, 1.4). Then find the
magnitude of AB .
10. __________________
11. Write EF as the sum of unit vectors for E(2.1, 2.6, 7)
and F(0.8, 7, 5).
11. __________________
Find each inner product and state whether the vectors are
perpendicular. Write yes or no.
13. 2, 6, 8 4, 2, 12
12. 8, 23 12, 6
12. __________________
13. __________________
Find each cross product.
14. 6, 12, 3 4, 2, 13
14. __________________
15. __________________
© Glencoe/McGraw-Hill
15.
14, 7,
4 347
5, 32, 2
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2A (continued)
16. Find the magnitude and direction of the
resultant vector for the diagram at the right.
16. __________________
17. What force is required to push a 147-pound crate up
a ramp that makes a 12° angle with the ground?
17. __________________
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
2, 3. Then write
point P 32, 5 and is parallel to a
parametric equations of the line and graph it.
19. __________________
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. _______________________________________________
23. x 2t 5; y 4t 74
24. Lisset throws a softball from a height of 4 meters, with
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?
24. __________________
25. A rectangular prism has vertices at A(1, 1, 3), B(1, 2, 3),
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.
25. __________________
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.
© Glencoe/McGraw-Hill
348
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2B
has a magnitude of 10 meters and a direction
1. The vector v
of 92°. Find the magnitude of its vertical and horizontal
components.
1. __________________
has a magnitude of 25.5 feet. If v
1u
, what 2. __________________
2. The vector u
3
?
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.
3b
3. a
3. __________________
b
4. 12a
4. __________________
5. Write the ordered pair that represents the vector from
A(0, 8) to B(1, 7). Then find the magnitude of AB.
5. __________________
of 27 newtons pulls at an angle of 23° above
6. A force F
1
of 33 newtons pulls at an angle of 55°
due east. A force F
2
below due west. Find the magnitude and direction of the
resultant force.
6. __________________
Find an ordered pair or ordered triple to represent u in
1, 6, w 2, 5, r 1, 1, 0, and
each equation if v
10, 6, 5.
s
v
3w
3s 2r
r 1s
8. u
9. u
7. u
5
7. __________________
8. __________________
9. __________________
10. Write the ordered triple that represents the vector from
A(5, 8, 9) to B(2, 2, 2). Then find the magnitude of AB.
10. __________________
11. Write EF as the sum of unit vectors for E(1, 2, 7) and
F(8, 7, 5).
11. __________________
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. __________________
© Glencoe/McGraw-Hill
349
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2B (continued)
16. Find the magnitude and direction of the resultant vector
for the diagram below.
16. __________________
17. Anita is riding a toboggan down a hill. If Anita weighs
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?
17. __________________
18. A 15-newton force acting at 30° and a 25-newton force
acting at 60° act concurrently on an object. What is the
magnitude and direction of a third force that produces
equilibrium on the object?
18. __________________
19. Write a vector equation of the line that passes through
3, 7. Then write
point P(1, 0) and is parallel to a
parametric equations of the line and graph it.
19. __________________
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
with the given parametric equations.
22. x t 6; y 2t 4
23. x 2t 5; y 4t 2
22. __________________
24. Pablo kicks a football with an initial velocity of 30 feet
per second at an angle of 58° with the horizontal. After
0.3 second, how far does the ball travel vertically?
24. __________________
23. __________________
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.
Bonus Write parametric equations for the line passing
through (2, 2) and parallel to the line with
equation 8x 2y 6.
© Glencoe/McGraw-Hill
350
25. __________________
Bonus: __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2C
has a magnitude of 5 meters and a direction
1. The vector v
of 60°. Find the magnitude of its vertical and horizontal
components.
1. __________________
has a magnitude of 4 centimeters.
2. The vector u
1
u
, what is the magnitude of v
?
If v
2. __________________
3
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. ab
3. ____________
b
4. 2a
4. ____________
5. Write the ordered pair that represents the vector from
.
A(3, 1) to B(1, 2). Then find the magnitude of AB
5. __________________
of 25 newtons pulls at an angle of 20° above
6. A force F
1
of 35 newtons pulls at an angle of 60°
due east. A force F
2
above due east. Find the magnitude and direction of the
resultant force.
6. __________________
Find an ordered pair or ordered triple to represent u in
1, 5, r 1, 1, 1,
each equation if v 2, 3, w
and s 0, 3, 2.
7. u
2v
w
8. u
s r
9. u
3s r
7. __________________
8. __________________
9. __________________
10. Write the ordered triple that represents the vector from
.
A(1, 3, 5) to B(3, 0, 1). Then find the magnitude of AB
10. __________________
11. Write EF as the sum of unit vectors for E(5, 1, 4) and
F(9, 3, 1).
11. __________________
Find each inner product and state whether the vectors
are perpendicular. Write yes or no.
12. __________________
12. 2, 0 0, 5
13. __________________
13. 3, 4, 2 2, 2, 1
14. __________________
Find each cross product.
14. 2, 1, 3 1, 0, 5
© Glencoe/McGraw-Hill
15. 2, 2, 1 0, 2, 2
351
15. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Test, Form 2C (continued)
16. Find the magnitude and direction
of the resultant vector for
the diagram at the right.
16. __________________
17. Matt is pushing a grocery cart on a level floor with a force
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?
17. __________________
18. A 10-newton force acting at 45° and a 20-newton force
acting at 130° act concurrently on an object. What is the
magnitude and direction of a third force that produces
equilibrium on the object?
18. __________________
19. Write a vector equation of the line that passes through
point P(3, 2) and is parallel to a 2, 6. Then write
parametric equations of the line and graph it.
19. __________________
y
O
x
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
45 feet per second at an angle of 12° with the horizontal.
After 0.5 second, what is the height of the ball?
24. __________________
25. A cube has vertices at A(2, 0, 0), B(2, 0, 2), C(2, 2, 2),
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.
25. __________________
Bonus
Write parametric equations for the line passing
through (0, 0) and parallel to 3y 9x 3.
© Glencoe/McGraw-Hill
352
Bonus: __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Open-Ended Assessment
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 the
requirements of the problem.
1. Given the vectors below, complete the questions that follow.
d 8, 11
c 3, 1, and .
a. Show two ways to find ab
. Explain each step.
b
b. Find a
b
c. Does ab
a? Why or why not?
b
b
d. Does a
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.
if a
1, 3, 0. Graph the vectors
b
2, 1, 0 and b
h. Find a
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.
© Glencoe/McGraw-Hill
353
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Mid-Chapter Test (Lessons 8-1 through 8-4)
has a magnitude of 12 inches and direction
1. The vector v
of 36°. Find the magnitude of its vertical and horizontal
components.
has a magnitude of 9.9 centimeters. If
2. The vector u
4u
, what is the magnitude of v
?
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.
2b
3. 2a
4. 3a
b
5. Write the ordered pair that represents the vector from
.
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)
and D(1, 2).
1. __________________
2. __________________
3. ____________
4. ____________
5. __________________
6. __________________
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
3, 8 and w 3, 4.
v
5w
2v
3w
4w
9. u
10. u
v
8. u
11. Write the ordered triple that represents the vector from
.
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
F(4, 2, 3).
8.
9.
10.
11.
Find an ordered triple to represent u in each equation if
0, 3, 4.
v 5, 2, 0, w 3, 8, 1, and x
13. __________________
14. __________________
15. u
x 12 v
15. __________________
v
w
13. u
3w
2x
14. u
__________________
__________________
__________________
__________________
12. __________________
Find each inner product and state whether the vectors
are perpendicular. Write yes or no.
16. __________________
16. 6, 4 2, 4
17. __________________
17. 4, 3, 1 8, 12, 4
18. __________________
Find each cross product.
18. 9, 1, 0 3, 2, 5
19. 6, 4, 2 1, 1, 3
20. Find a vector that is perpendicular to both c 0, 3, 6
4, 2, 5.
and d
© Glencoe/McGraw-Hill
354
19. __________________
20. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8, Quiz A (Lessons 8-1 and 8-2)
1. The vector v
has a magnitude of 13 millimeters and a
direction of 84°. Find the magnitude of its vertical and
horizontal components.
2a
has a magnitude of 6.3 meters. If b
,
2. The vector a
what is the magnitude of b ?
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.
b
3. 2a
2b
4. a
1. __________________
2. __________________
3. __________________
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
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.
7. __________________
Find an ordered pair to represent u in each equation if
8. __________________
3, 4.
v 6, 6 and w
9. __________________
5w
8. u
3w
9. u 2v
4w
v
10. u
10. _____________
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8, Quiz B (Lessons 8-3 and 8-4)
1. Write the ordered triple that represents the vector from
.
A(3, 4, 10) to B(8, 4, 2). Then find the magnitude of AB
1. __________________
2. Write EF as a sum of unit vectors for E(8, 2, 4) and
F(5, 3, 0).
2. __________________
1 w
z if
3. Find an ordered triple that represents 2v
3
2, 1, 5, w
3, 4, 6, and z 0, 3, 2.
v
3. __________________
if a
7, 3, 8 and
4. Find the inner product of a and b
5, 2, 4. Are perpendicular?
a and b
b
4. __________________
and 5. Find the cross product of c
d if c 5, 5, 4 and
d 2, 3, 6. Verify that the resulting vector is
.
perpendicular to c and d
5. __________________
© Glencoe/McGraw-Hill
355
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8, Quiz C (Lessons 8-5 and 8-6)
1. Find the magnitude and direction
of the resultant vector for
the figure at the right.
1. __________________
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. __________________
y
2, 4. Then write parametric
P(1, 3) and is parallel to q
equations of the line and graph it.
O
Write parametric equations for each equation.
5. 6x y 2
6. 2x 5y 4
x
5. __________________
6. __________________
Write an equation in slope-intercept form of the line
with the given parametric equations.
7. x 6t 8
8. x 3t 10
y t 4
y 4t 2
7. __________________
8. __________________
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8, Quiz D (Lessons 8-7 and 8-8)
While positioned 25 yards directly in front of the goalposts, Bill kicks the football with
an 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
5. the dimensions are increased by a factor of 3
© Glencoe/McGraw-Hill
356
4. __________________
5. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 SAT and ACT Practice
6. In ABC, A is a right angle. If
BC 25 and AB 20, which is the
area of ABC?
A 187.5 units2
B 250 units2
C 753
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 is
NOT true for the diagram below?
After working each problem, record the
correct answer on the answer sheet
provided or use your own paper.
Multiple Choice
1. If the area of a circle is 49, what
is the circumference of the circle?
A 7
B 7
C 14
D 14
E 49
2. If all angles in the figure below are
right 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 53
B 2x 16y 53
C 2x 16y 43
D 2x 16y 43
E None of these
4. If A(0, 0) and B(8, 4) are vertices of
ABC and ABC is isosceles, what
are 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 of
five rectangular solids. All have the
same volume EXCEPT
A 8 by 6 by 5
B 4 by 15 by 2
C 15 by 15 by 40
A m6 m9
B m3 m6 90°
C m2 m6 m5 180°
D m8 m2 m3
E m4 m2 m9
9. If 2 y 50 and y 2x 1, then which
of the following statements is true?
A x 13
B 16.5 x 32.5
C 2 x 2.5
D 3 x 3.5
E 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 6
B x 3
C x 6
D x 3
E None of these
D 13 by 24 by 15
E 12 by 4 by 60
© Glencoe/McGraw-Hill
357
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 SAT and ACT Practice (continued)
11. A circle is inscribed in a square as
shown in the figure below. What is the
ratio of the area of the shaded region
to the area of the square?
A 4
1 B 4
4 C 4
4
D 4
E 1
17–18. Quantitative Comparison
A if the quantity in Column A is
greater
B if the quantity in Column B is
greater
C if the two quantities are equal
D if the relationship cannot be
determined from the information
given
Column A
17. Square X has sides of length x.
Square Y has sides of length 2x.
12. Each angle in the figure below is a
right angle. Find the perimeter of the
figure.
7
A 11 units
B 18 units
4
C 22 units
D 24 units
E 28 units
Area of square X
Half the area
of square Y
18. ABCD is a rectangle.
E
B
13. Which number is 45 of 34 of 10?
A 6
B 4
C 3
D 1.5
E 0.5
C
D
A
Area of DBC
14. Evaluate 9[42(2)4 32]1.
A 8
B 18
Area of AED
19. Grid-In BDE is contained in
rectangle ABCD as shown below. Find
the area of BDE in square units.
C 18
D 8
E None of these
15. A solid cube has 4-inch sides. How
many straight cuts through the cube
are needed to produce 512 small cubes
that have half-inch sides?
A 7
B 9
C 16
D 21
E None of these
16. A roll of wallpaper is 15 inches wide
and can cover 39 square feet. How long
is the roll?
A 2.6 ft. B 21.7 ft.
C 31.2 ft D 46.9 ft.
E None of these
© Glencoe/McGraw-Hill
Column B
358
D
9
C
4
A
3
E
B
20. Grid-In The area of a rhombus is
28 square units. The length of one
diagonal is 7 units. What is the length
of the other diagonal in units?
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Chapter
8
Chapter 8 Cumulative Review (Chapters 1-8)
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).
Find the image of the triangle after a reflection over
the x-axis.
3. __________________
4. Find the inverse of
3 1
if it exists. If it does not
5 0
4. __________________
exist, write none.
5. Write the equation obtained when ƒ(x) x is
translated 3 units down and compressed
horizontally by a factor of 0.5.
5. __________________
6. Solve x 3
6. __________________
5.
7. Determine the rational roots of 2x3 3x2 17x 12 0.
7. __________________
1 2
8. Solve x
x1
8. __________________
0.
9. Identify all angles that are coterminal with a 232 angle.
Then find one positive angle and one negative angle
coterminal with the given angle.
9. __________________
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. __________________
,
5
13. If is an angle in the second quadrant and cos 3
find tan 2.
13. __________________
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
whose parametric equations are x 3 7t and y 4 t.
© Glencoe/McGraw-Hill
359
15. __________________
Advanced Mathematical Concepts
Blank
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Review, Chapters 5-8
Find the value of the given trigonometric
function for angle in standard position
if a point with the given coordinates lies
on its terminal side.
1. cos ; (2, 3)
2. tan ; (10, 2)
3. sin ; (4, 1) 4. sec ; (1, 0)
Solve each problem. Round to the
nearest tenth.
Solve.
20. Given a central angle of 60°, find the
length of its intercepted arc in a circle
of radius 6 inches. Round to the
nearest tenth.
Find each value by referring to the graph
of the sine or the cosine function.
21. sin 22. cos 2
7
23. sin 2
State the amplitude and period for each
function.
25. y 2 cos 3x
26. y 5 tan 5x
27. y 4 cot 2x 2
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.
Graph each function.
28. y 12 cos 2x
29. y 3 tan 2x 2
30. y x 2 sin 3x
Evaluate each expression.
8. cos Arccos 14
9. cot Cos1 23
10. cos (Sin1 0) sin (Tan1 0)
Write the equation for the inverse of
each relation. Then graph the relation
and its inverse.
31. y arccos x
32. y cot x
Determine the number of possible
solutions for each triangle. If a solution
exists, solve the triangle. Round to the
nearest tenth.
11. A 46°, a 86, c 200
12. a 19; b 20, A 65°
13. A 73°; B 65°, b 38
Use the given information to determine
each trigonometric value.
33. sec 43, 0° 90°; cos Find the area of each triangle. Round to
the nearest tenth.
14. a 5, b 9, c 6
15. a 22, A 63°, B 17°
Change each radian measure to degree
measure.
3
17. 16. 2
4
7
18. 2
© Glencoe/McGraw-Hill
24. cos (6)
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 x
37. sin (180° ) tan cos Use sum or difference identities to find
the exact value of each trigonometric
function.
38. sin 105°
39. cos 135°
40. tan 15°
41. sin (210°)
7
19. 12
361
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Review, Chapters 5-8 (continued)
Find an ordered pair to represent a in
each equation if b 1, 3 and
c 2, 2.
c
58. a b c
57. a b 59. a 3b 2c
60. a 3b c
If x is an angle in the first quadrant and
sin x 25 , find each value.
42. cos 2 x
44. tan x2
43. sin x2
45. sin 2x
Solve each equation for 0° x 180°.
46. sin2 x sin x 0
47. cos 2 x 4 cos x 3
48. 5 cos x 1 3 cos 2x
Find an ordered triple to represent u in
each equation if v 3, 1, 1 and
5, 2, 3. Then write u as the sum
w
of unit vectors.
2v
w
v
2w
62. u
61. u
3v
3w
4v
2w
63. u
64. u
Write each equation in normal form.
Then find the length of the normal and
the angle that it makes with the
positive x-axis.
49. 2x 3y 2 0
50. 5x 2y 8
51. y 3x 7
Find the distance between the point with
the given coordinates and the line with
the given equation.
52. (2, 5); 2 x 2y 3 0
53. (2, 2); x 4y 6
54. (1, 3); 4x y 1 0
for Exercises 55-56.
and b
Use vectors a
Find each inner product or cross
product.
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 that
passes through point P and is parallel
to v. Then write parametric equations of
the line.
1, 5
68. P(0, 5), v
2, 2
69. P(4, 3), v
55. Use a ruler and a protractor to
determine the magnitude (in
centimeters) and direction of the
.
resultant ab
56. Find the magnitude of the vertical and
.
horizontal components of a
© Glencoe/McGraw-Hill
362
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Test, Chapters 5-8
1. True or false: sin (85°) sin 85°.
1. __________________
2. Find the area of ABC if a 12, b 15, and c 23. Round
to the nearest square unit.
2. __________________
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
intercepted arc in a circle of radius 15 inches. Round
to the nearest tenth.
5. __________________
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.
5
.
8. Find the value of Sin1 sin 6
7. __________________
8. __________________
9. Use the Law of Sines to solve ABC when a 1.43,
b 4.21, and A 30.4°. If no solution exists, write none.
9. __________________
10. Use the sum or difference identity to find the exact value
of tan 105°.
10. __________________
11. Find the distance between P(7, 4) and the line with
equation x 3y 5 0. Round to the nearest tenth.
11. __________________
12. Find the inner product of the vectors 2, 5 and 4, 2.
Then state whether the vectors are perpendicular.
Write yes or no.
12. __________________
© Glencoe/McGraw-Hill
363
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Test, Chapters 5-8 (continued)
13. Find the value of sin for angle in standard position if a
point with coordinates (3, 2) lies on its terminal side.
13. __________________
14. Solve sin 1 for all real values of .
14. __________________
15. A car’s flywheel has a timing mark on its outer edge.
The height of the timing mark on the rotating flywheel
. Graph one full cycle
is given by y 3.55 sin x 4
of this function.
15.
if w
6, 4.
16. Find the ordered pair that represents 3 w
16. __________________
as the sum of unit vectors for X(8, 2, 9) and
17. Write XY
Y(12, 1, 10).
17. __________________
18. In the triangle at the right, b 6.2
and c 8.2. Find to the
nearest tenth.
18. __________________
19. If 0°
,
find cos .
3
90° and tan 2
20. Solve sin2 x sin x 2 0 for 0°
x
360°.
19. __________________
20. __________________
21. If 849° is in standard position, determine a coterminal
angle that is between 0° and 360°. State the quadrant in
which the terminal side lies.
21. __________________
tan x csc x 1 is an identity. Write your
22. Verify that sec x
answer on a separate piece of paper.
22. __________________
© Glencoe/McGraw-Hill
364
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Test, Chapters 5-8 (continued)
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
.
x the interval 2
2
26.
7
radians to degree measure.
27. Change 9
27. __________________
28. Nathaniel pulls a sled along level ground with a force of
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.
28. __________________
29. State the amplitude, period, and phase shift of the
function y 2 sin (4 2).
29. __________________
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,
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.
© Glencoe/McGraw-Hill
365
31. __________________
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
UNIT
2
Unit 2 Test, Chapters 5-8 (continued)
32. Find the value of Tan1 1.
32. __________________
33. Use the Law of Cosines to solve ABC with A 126.3°,
b 45, and c 62.5. Round to the nearest tenth.
33. __________________
34. Write an equation in slope-intercept form of the line with
parametric equations x 2 3t and y 4 t.
34. __________________
35. Verify that cos (90° A) sin A is an identity.
35. __________________
36. Write the equation for the inverse of the function
y Cos x. Then graph the function and its inverse.
36. __________________
37. Find sin (Sin1 14).
37. __________________
38. Find the area of a sector if the central angle measures
38. __________________
3
5
6
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
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?
40. __________________
© Glencoe/McGraw-Hill
366
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(10 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
© Glencoe/McGraw-Hill
A1
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
SAT and ACT Practice Answer Sheet
(20 Questions)
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
© Glencoe/McGraw-Hill
A2
Advanced Mathematical Concepts
© Glencoe/McGraw-Hill
2 cm; 60°
2.
3 cm; 140°
A3
7. y 3z
0.6 cm; 217°
2.5 cm; 83°
3.
1cm; 310°
2.30 cm, 1.93 cm
9. y
0.64 cm, 0.77 cm
10. z
© Glencoe/McGraw-Hill
318
Advanced Mathematical Concepts
11. Aviation An airplane is flying at a velocity of 500 miles per hour
due north when it encounters a wind blowing out of the west at
50 miles per hour. What is the magnitude of the airplane's resultant
velocity? 502.49 mph
1.00 cm, 1.73 cm
8. x
Find the magnitude of the horizontal and vertical components of
each vector shown in Exercises 1-3.
5.4 cm; 93°
6. 2x
y
3.9 cm; 110°
Find the magnitude and direction of each resultant.
y
5. x z
4. x 1.
Now add the resulting vector
to the 120-N vector.
The resultant force is 219 N, with an amplitude of 145°.
Add the 80-N vector and
the 100-N vector first.
Add two vectors at a time. The order in which the vectors
are added does not matter.
A force of 80 N acts on an object at an
angle of 70° at the same time that a
force of 100 N acts at an angle of 150°.
A third force of 120 N acts at an angle
of 180°. Find the magnitude and
direction of the resultant force acting on
the object.
© Glencoe/McGraw-Hill
1. One force acts with 40 N at 50° on
an object. A second force acts with
100 N at 110°. A third force acts with
10 N at 150°. Find the magnitude
and amplitude of the resultant
force. 131 N; 98°
319
Advanced Mathematical Concepts
176 N; 112°
2. One force acts with 75 N at 45°. A
second force acts with 90 N at 90°.
A third force acts with 120 N at 170°.
Find the magnitude and amplitude
of the resultant force.
Find the magnitude and amplitude of the resultant force acting on each object.
Example
Three or more forces may work on an object at one time. Each of these
forces can be represented by a vector. To find the resultant vector that
acts upon the object, you can add the individual vectors two at a time.
Enrichment
Use a ruler and a protractor to determine the magnitude
(in centimeters) and direction of each vector.
8-1
More Than Two Forces Acting on an Object
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Geometric Vectors
8-1
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 8-1)
Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
1, 3; 兹1
苶0
苶
2. A(4, 2), B(5, 5)
A4
21, 28
3v
7. u 5w
7, 7
11, 5; 兹1
苶4
苶6
苶
3. A(3, 6), B(8, 1)
兹4
苶1
苶; 4 i 5 j
© Glencoe/McGraw-Hill
295.62 N
321
Advanced Mathematical Concepts
10. Gardening Nancy and Harry are lifting a stone statue and
moving it to a new location in their garden. Nancy is pushing the
statue with a force of 120 newtons (N) at a 60° angle with the
horizontal 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?
2兹1
苶0
苶; 2 i 6 j
Find the magnitude of each vector, and write each vector as the
sum of unit vectors.
8. 2, 6
9. 4, 5
1, 11
3w
6. u 4v
6, 3
Find an ordered pair to represent u in each equation if
w 3, 5.
v 2, 1 and 5. u
w 2v
4. u 3v
3, 1; 兹1
苶0
苶
1. A(2, 4), B(1, 3)
Write the ordered pair that represents AB . Then find the
magnitude of AB.
Algebraic Vectors
8-2
2
3
1
–4
2r s
3r 4s
2
5
冣
1
4
16
3
1
5
–1
_1_
2
1
0
4
_1_
2
2
–2
© Glencoe/McGraw-Hill
1
冢 冣, u៝ 冢 冣, w៝ 冢 冣
19
w
2
–3
4
v
–
u
w
2
3. v
៝
v
6
u
៝
1. v
冢 冣, u៝ 冢 冣, w៝ 冢 冣
៝ –
u
៝
r –
and s – . So, v
11
11
11
3
322
1
–1
2
3
16
u
5
2
–7
4
2
13
w
10
Advanced Mathematical Concepts
–1
–3
1
冢 冣, u៝ 冢 冣, w៝ 冢 冣
v
4. v
៝
v
u
4
w
2. v
៝
冢 冣, u៝ 冢 冣, w៝ 冢 冣
16
w
៝.
11
Solving the system of equations yields the solution
–2 2r s
5 3r 4s
–2
5
冢 冣 r冢 冣 s冢 冣 冢
2
3
Write the vector v
as a linear combination of
៝ and w
៝ .
the vectors u
Every vector v៝ v2 can be written as a
linear combination of any two nonparallel vectors
u
៝ and w
៝. The vectors u
៝ and w
៝ are said to form a
basis for the vector space v2 which contains
all vectors having 1 column and 2 rows.
Write each vector as a linear combination of the vectors u and w.
Example
Linear Combination
Theorem in v2
The expression v r៝
u s៝
w, the sum of two vectors each multiplied by
៝ and w
៝.
scalars, is called a linear combination of the vectors u
_1_
4
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Basis Vectors
8-2
Answers
(Lesson 8-2)
Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
2兹1
苶4
苶
A5
1, 11, 17; 兹4
苶1
苶1
苶
© Glencoe/McGraw-Hill
324
Advanced Mathematical Concepts
11. Physics Suppose that the force acting on an object can be
expressed by the vector 具85, 35, 110典, where each measure in
the ordered triple represents the force in pounds. What is the
magnitude of this force? 143 lb
2, 4, 7
2w
10. u 5v
3w
9. u 4v
26, 40, 47
4, 4, 4
8. u v w
8, 12, 14
7. u v w
Find an ordered triple to represent u in each equation if
6, 8, 9.
v 2, 4, 5 and w
11, 3, 17; 兹4
苶1
苶9
苶
6. A(6, 8, 5), B(7, 3, 12)
3, 1, 9; 兹9
苶1
苶
6, 4, 4; 2兹1
苶7
苶
5. A(4, 5, 8), B(7, 2, 9)
4. A(4, 0, 6), B(7, 1, 3)
3. A(2, 1, 3), B(4, 5, 7)
Write the ordered triple that represents AB . Then find the
magnitude of AB .
兹1
苶0
苶1
苶
Locate point B in space. Then find the magnitude of a vector from
the origin to B.
1. B(4, 7, 6)
2. B(4, 2, 6)
Vectors in Three-Dimensional Space
8-3
NAME _____________________________ DATE _______________ PERIOD ________
r
1
3
1
s
1
–2
1
1
3
1
t
–1
–1
1
, w
៝
1
2
1
rst
3r 2s t
rst
冣
, and z
៝
1
1
1
.
冢冣
as a linear combination of
Solving the system of equations yields the solution
r 0, s 1, and t 2. So, v
៝ w
៝ 2៝
z.
–1 r s t
– 4 3r 2s t
3rst
–1
–4
3
the vectors u
៝
1
4
3
冢 冣
冢冣 冢 冣
冢冣 冢冣 冢冣 冢冣 冢
៝
Write the vector v
–6
–2
2
, u
៝
1
1
0
, w
៝
, w
៝
23
w
7
1
–2
3
–1
0
1
1
–1
2
1
, u
៝
© Glencoe/McGraw-Hill
, w
៝
3
z
2
1
2
–1
, and z
៝
, and z
៝
1
z
7
2
2
1
325
1
0
1
冢冣
4
2
–1
0
1
1
冢冣
冢冣
, and z
៝
冢冣 冢 冣 冢冣
v
–
u
2
3. v
៝
,u
៝
8
u
7
5
–2
0
冢冣 冢冣 冢冣
v
2. v
៝
1
0
1
冢 冣 冢冣 冢冣
v
–5
u
w 3
z
៝
1. v
Advanced Mathematical Concepts
៝, w
៝, and z៝.
Write each vector as a linear combination of the vectors u
Example
៝ v3 can be written as a linear combination of any
Every vector v
three nonparallel vectors. The three nonparallel vectors, which must
be linearly independent, are said to form a basis for v3, which contains
all vectors having 1 column and 3 rows.
The expression v
៝ r៝
u s៝
w t៝
z, the sum of three vectors each
៝, w
៝,
multiplied by scalars, is called a linear combination of the vectors u
៝.
and z
Enrichment
Basis Vectors in Three-Dimensional Space
8-3
Answers
(Lesson 8-3)
Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
2; no
3; no
0; yes
5. 具4, 1, 1典 具1, 3, 4典 6. 具0, 0, 1典 具1, 2, 0典
11; no
A6
© Glencoe/McGraw-Hill
300 ft-lb
327
Advanced Mathematical Concepts
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 the
horizontal. How much work is Janna doing in the vertical
d.)
direction? (Hint: Use the sine ratio and the formula W F
12, 18, 0; yes
12. 具0, 0, 6典 具3, 2, 4典
7, 0, 14; yes
11. 具6, 1, 3典 具2, 2, 1典
具4, 1, 0典 具5, 3, 1典
1, 4, 7; yes
10.
27, 3, 14; yes
7, 1, 11; yes
9. 具3, 1, 2典 具2, 3, 1典
3, 3, 3; yes
Find each cross product. Then verify that the resulting vector is
perpendicular to the given vectors.
7. 具1, 3, 4典 具1, 0, 1典
8. 具3, 1, 6典 具2, 4, 3典
9; no
4. 具2, 0, 1典 具3, 2, 3典
0; yes
Find each inner product and state whether the vectors are
perpendicular. Write yes or no.
1. 具3, 6典 具4, 2典
2. 具1, 4典 具3, 2典
3. 具2, 0典 具1, 1典
Perpendicular Vectors
8-4
4c
a
4b
–2
c
a
2b
a
–1
0
c
a
2b
1
2
4c
a
4b
–2
–1
具12,–3典 具4, 1典
0
1
具0, 1典 具0,–3典
2
具4,–11典
c
៝ 具0, 1, 2典
c
៝ 具1, 1, 0典
© Glencoe/McGraw-Hill
328
3៝
4. f (x) x3a
៝ xb
c
具1, –2, 0典 c
a
៝ 具0,1, –2典 b
៝ 具–2, 0, 1典
3. f (x) x 2 c
៝ 3xa
៝ 4b
具3, 2, 1典
a
៝ 具1, 1, 1典 b
5៝
2. f (x) 2x 2a
៝ 3x b
c
具1, 0, 1典
a
៝ 具0, 1, 1典 b
– 5, –3,5
–6, 0, 3
–7, 3, 1
–9, 33, –51
f (x)
–12, – 8, –4
–9, –4, 1
–6, 2, 10
–3, 10, 23
f (x)
–11, 3, 2
–8, –3,–1
–5, –5, 0
–2, –3, 5
f (x)
–14, –4
0, 0
6, – 8
10, –22
Advanced Mathematical Concepts
1
3
–1
0
x
2
3
0
1
x
0
1
–2
–1
x
For each of the following, complete the table of resulting vectors.
3x៝
៝ 2x2 b
c
1. f (x) x 3a
x
具2, 3典 c
a
៝ 具1, 1典 b
៝ 具3, –1典
–1
0
1
2
f (x)
x
具1, 1典, and c
៝ 具0, 1典, b
៝ 具2, –2典, the resulting vectors for the
If a
values of x are as follows.
f (x)
x
and c
Let a
៝ , b,
៝ be fixed vectors. The equation f (x) a
៝ 2x b x 2 c
៝
defines a vector function of x. For the values of x shown, the
assigned vectors are given below.
f (x)
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
Vector Equations
8-4
Answers
(Lesson 8-4)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
A7
11.39 N; 50.74°
© Glencoe/McGraw-Hill
249.80 km; 16.10° south of east
330
Advanced Mathematical Concepts
7. Aviation A pilot flies a plane east for 200 kilometers, then 60°
south of east for 80 kilometers. Find the plane's distance and
direction from the starting point.
6. Nadia is pulling a tarp along level ground with a force of 25
pounds directed along the tarp. If the tarp makes an angle of 50°
with the ground, find the horizontal and vertical components of
the force. 16.07 lb; 19.15 lb
5. What would be the force required to push a 200-pound object up a
ramp inclined at 30° with the ground? at least 100 lb
281.78 N; 27.47°
Find the magnitude and direction of the resultant vector for each
diagram.
3.
4.
2. a force of 85 pounds due north and a force of 100 pounds due west
acting on the same object
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
Applications with Vectors
8-5
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment
1
.
2
具–1, 2, 1典 具1, –1, 2典 具0, –2, –6典 具0, 0, 0典, so the
three vectors are linearly dependent.
Hence, one solution is a 1, b 1, and c The above system does not have a unique solution. Any
solution must satisfy the conditions that a b 2c.
–a b 0
2a b 2c 0
a 2b 6c 0
Solve a具–1, 2, 1典 b具1, –1, 2典 c具0, –2, –6典 具0, 0, 0典.
Are the vectors 具–1, 2, 1典, 具1, –1, 2典, and 具0, –2, –6典
linearly dependent?
© Glencoe/McGraw-Hill
no
5. 具2, –4, 6典, 具3, –1, 2典, 具–6, 8, 10典
no
3. 具1, 1, 1典, 具–1, 0, 1典, 具1, –1, –1典
yes; –2, 6 21, –3 0, 0
1. 具–2, 6典, 具1, –3典
331
no
9
4
典
Advanced Mathematical Concepts
具
6. 具1, –2, 0典, 具2, 0, 3典, –1, 1,
yes; 21, 1, 1 –1, 0, 1 –3, –2, –1 0, 0, 0
4. 具1, 1, 1典, 具–1, 0, 1典, 具–3, –2, –1典
yes; 23, 6 32, 4 0, 0
2. 具3, 6典, 具2, 4典
Determine whether the given vectors are linearly dependent. Write yes or no. If the
answer is yes, give a linear combination that yields a zero vector.
Example
A set of vectors is called linearly dependent if and only if there
exist scalars, not all zero, such that a linear combination of the
vectors yields a zero vector.
The zero vector is 具0, 0典 in two dimensions, and 具0, 0, 0典 in three
dimensions.
Linearly Dependent Vectors
8-5
NAME _____________________________ DATE _______________ PERIOD ________
Answers
(Lesson 8-5)
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
x 5, y 8 t9, 2
x 5 9t
y 8 2t
4. P(5, 8), a 具9, 2典
x 3, y 7 t4, 5
x 3 4t
y 7 5t
A8
y 3x 15
© Glencoe/McGraw-Hill
333
Advanced Mathematical Concepts
Chad x 1, y 19 t; Brett x 12 11t, y 19t; yes
11. Physical Education Brett and Chad are playing touch football
in gym class. Brett has to tag Chad before he reaches a 20-yard
marker. Chad follows a path defined by 具x 1, y 19典 t具0, 1典,
and Brett follows a path defined by 具x 12, y 0典 t具11, 19典.
Write parametric equations for the paths of Brett and Chad. Will
Brett tag Chad before he reaches the 20-yard marker?
y 12x 121
Write an equation in slope-intercept form of the line with the
given parametric equations.
9. x 2t 3
10. x t 5
yt4
y 3t
xt
y 54t 5
8. 5x 4y 20
7. 3x 2y 6
xt
y 32t 3
xt
y t 4
xt
y 3t 8
Write parametric equations of the line with the given equation.
5. y 3x 8
6. y x 4
x 2, y 4 t 1, 3
x2t
y 4 3t
3. P(2, 4), a 具1, 3典
x 2, y 1 t3, 4
x 2 3t
y 1 4t
Write a vector equation of the line that passes through point P
and is parallel to a. Then write parametric equations of the line.
2. P(3, 7), a 具4, 5典
1. P(2, 1), a 具3, 4典
Practice
Vectors and Parametric Equations
8-6
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
y1
2
z0
–1
y1
,
2
and t =
z0
.
–1
x+1
,
1
1
1
2
1
1
冣
d
1 12
2
1
2
1, 2
1
2
1
2
2
⬇ 1.2 units
冢 冣 1, 冣
(1 2)2 0 12
冢
冪冢莦莦莦莦冢 莦冣莦冣 莦莦莦莦莦莦莦冢 莦莦冢 莦冣莦冣
Use the distance formula:
, 2, .
or 2
2
冢
The point of intersection is
intersection of the vector and the plane.
This means that t at the point of
2
t
Substitute these values into the equation of the plane.
(t 1) 2(2t 1) (–t) 4
Solve for t: 6t 1 4
So, x t 1, y 2t 1, and z –t are parametric
equations of the line.
t=
Set t equal to each of the above ratios. Then, t =
The denominators of these ratios represent a vector that
is perpendicular to the plane, and passes through the
given point.
x1
1
Use the coefficients of the equation of the plane and the
coordinates of the point to write the ratios below.
Find the distance from P(1, 1, 0) to the plane
x 2y z 4.
© Glencoe/McGraw-Hill
0.8 unit
1. from (2, 0, –1) to x 2y z 3
334
0.3 unit
Advanced Mathematical Concepts
2. from (1, 1, –1) to 2x y 3z 5
Find the distance from the given point to the given plane. Round your answers to the
nearest tenth.
Example
You can use parametric equations to help you find the distance from a
point not on a plane to a given plane.
Using Parametric Equations to Find the
Distance from a Point to a Plane
8-6
Answers
(Lesson 8-6)
Advanced Mathematical Concepts
Practice
© Glencoe/McGraw-Hill
A9
© Glencoe/McGraw-Hill
22.98 ft; 23.98 ft
336
Advanced Mathematical Concepts
b. After 0.5 seconds, how far has the javelin traveled horizontally
and vertically?
x 65t cos 45°; y 65t sin 45° 16t2 5
6. Sports Jessica throws a javelin from a height of 5 feet with an
initial velocity of 65 feet per second at an angle of 45° with the
ground.
a. Write parametric equations to represent the path of the
javelin.
0.72 s; 48.43 ft
b. How long will it take the rocket to travel 10 feet horizontally
from its starting point? What will be its vertical distance at
that point?
x 80t cos 80°; y 80t sin 80° 16t2
5. Model Rocketry Manuel launches a toy rocket from ground
level with an initial velocity of 80 feet per second at an angle of
80° with the horizontal.
a. Write parametric equations to represent the path of the rocket.
39.85 ft/s, 93.89 ft/s
4. a golf ball hit with an initial velocity of 102 feet per second at an
angle of 67° with the horizontal
5.30 yd/s, 8.48 yd/s
3. a football thrown at a velocity of 10 yards per second at an angle
of 58° with the ground
6.59 ft/s, 62.65 ft/s
2. a toy rocket launched from level ground with an initial velocity of
63 feet per second at an angle of 84° with the horizontal
28.05 ft/s, 27.09 ft/s
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
Modeling Motion Using Parametric Equations
8-7
NAME _____________________________ DATE _______________ PERIOD ________
NAME _____________________________ DATE _______________ PERIOD ________
冣x
g
2v02 cos 2␣
冢
2
(tan ␣)x,
冢
9.8
2(120)2 cos 2 10°
冣x
2
y – 0.00035x2 0.18x
y–
(tan 10°)x
© Glencoe/McGraw-Hill
y – 0.013x 2 5.67x
1. a projectile fired at 80° to the
horizontal with an initial velocity
of 200 ft/s
337
Advanced Mathematical Concepts
2. a projectile fired at 40° to the
horizontal with an initial velocity
of 150 m/s
y – 0.00037x 2 0.84x
Write the equation of a projectile fired at an angle
of 10° to the horizontal with an initial velocity of
120 m/s.
Find the equation of the path of each projectile.
Example
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.
y–
The equation of the path of a projectile on the coordinate plane is
given by,
The path assumes that gravity is the only force acting on the
projectile.
The path of a projectile after it is launched is a parabola when graphed
on a coordinate plane.
Enrichment
Coordinate Equations of Projectiles
8-7
Answers
(Lesson 8-7)
Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill
冥
2.
冤
1
1 1
1
1
1 1
1
1 1 1 1
冥
0
0
0
A10
冥
冤
1 1 1 1
1
4
2
2
4
4
0
0 2
0 2
© Glencoe/McGraw-Hill
339
1
1
2
4
0 2
冥
Advanced Mathematical Concepts
reflection over xy-plane
dimensions increased
by a factor of 2
冥
冤
0 0
1 0
0 1
冤 冥
1
0
0
each matrix. Graph each
1
2
2
Transform the figure in Question 2 using
image and describe the result.
6.
5. 2 0 0
0 2 0
0 0 2
冤
1 3 1 1 1 3 3 3
2 4 2 4 4 2 2 4
0 2 2 0 2 0 2 0
Translate the figure in Question 1 using the given vectors. Graph each image and
write the translated matrix.
4. b 具1, 2, 2典
3. a 具1, 2, 0典
0 2 0 0 0 2 2 2
0 2 0 2 2 0 0 2
0 2 2 0 2 0 2 0
冤
Write the matrix for each figure.
1.
Practice
Transformation Matrices in Three-Dimensional Space
8-8
NAME _____________________________ DATE _______________ PERIOD ________
2
© Glencoe/McGraw-Hill
2
2
c
a
b
a2 b2 c2
340
Advanced Mathematical Concepts
冢兹a苶苶苶苶b苶苶苶c苶, arctan 冢 冣, arccos 冢 兹苶苶苶苶苶苶苶苶苶苶 冣冣
8. (a, b, c)
(3.7, 27°, 143°)
7. (–2, –3, –1)
(13, 37°, 23°)
6. (4, 12, 3)
Find the spherical coordinates of the point with the given rectangular coordinates.
Round distances to the nearest tenth and angles to the nearest degree.
(兹6
苶5
苶, 72°, 52°)
5. Use inverse trigonometric functions to find ␪ and ␾ to the nearest
degree. Write the spherical coordinates of Q.
兹6
苶5
苶
4. Find d.
5
3. Find QB.
2兹1
苶0
苶
2. Find OB by using the Pythagorean theorem.
2; 6
1. Find OA and AB.
The figure at the right shows point Q with rectangular coordinates (2, 5, 6).
Point P(d, ␪, ␾) in three-dimensional space is located using
three spherical coordinates:
d distance from origin
␪ angle relative to x -axis
␾ angle relative to y-axis
Note that the orientation of the axes shown is a different
perspective than that used in your textbook.
There are many coordinate systems for locating a point in the
two-dimensional plane. You have studied one of the most common systems, rectangular coordinates. The most commonly
used three-dimensional coordinate systems are the extended
rectangular system, with an added z-axis, and the spherical
coordinate system, a modification of polar coordinates.
Enrichment
Spherical Coordinates
8-8
Answers
(Lesson 8-8)
Advanced Mathematical Concepts
Chapter 8 Answer Key
Form 1A
Page 341
1.
B
2.
B
Form 1B
Page 342
12.
13.
3.
4.
5.
6.
A
B
A
8.
C
11.
1.
A
2.
C
B
12.
D
13.
C
14.
3.
A
4.
C
5.
B
B
14.
D
15.
B
16.
A
17.
D
18.
C
19.
B
20.
B
D
7.
10.
C
Page 344
C
15.
A
6.
16.
9.
Page 343
D
17.
B
18.
C
A
D
19.
B
20.
B
B
© Glencoe/McGraw-Hill
7.
D
8.
D
9.
B
10.
A
11.
Bonus:
B
B
Bonus:
A
A11
A
Advanced Mathematical Concepts
Chapter 8 Answer Key
Form 1C
Page 345
1.
B
2.
C
Form 2A
Page 346
12.
B
Page 348
1. 10.57 m, 4.27 m
16.
1 N, 313
17.
30.6 lb
313.6 in.
2.
13.
3.
Page 347
C
A
18. 26.6 N, 231.5
14.
4.
A
3. 4.2 cm, 43
C
x 32, y 5 t 2, 3;
19.
x 32 2t,
y 5 3t
4. 0.1 cm, 223
5.
C
15.
6.
18.5 N,
6. 2.2 above east
B
16.
7.
5. 1.9, 8.9; 9.10
D
B
21. x t, y 6t 15
D
23, 34
7.
8.
A
9.
C
10.
A
20. x t, y 43t 3
17.
C
18.
A
8
1 , 19
3, 4
8. 2
8
1
1, 1
3, 1
1
9. 3
4
2
8.9, 7.8, 10.4;
22.
y 4x 20
23.
3
y 2x 3
24.after 2.1 seconds
10. 15.75
19.
D
11. 2.9i 4.4 j 2k
25.
20.
11.
A
12.
0; yes
13.
8; no
14.
15.
365 , 14, 14
20, 42,1 2787 C
Bonus:
© Glencoe/McGraw-Hill
B
A12
4
A (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),
H (1, 3, 0)
Bonus:
x t,
y 1 t 5
2
12
Advanced Mathematical Concepts
Chapter 8 Answer Key
Form 2B
Page 350
Page 349
1. 9.99 m, 0.35 m
2.
Form 2C
Page 351
16. 13.7 N; 65.4
1. 4.33 m, 2.5 m
2.
8.5 ft
17.
4
3
16.
cm
17. 7.0 N, 13.2 N
3.
3.9 cm, 49
18. 23.1 N, 284.5
18. 38.7 N, 228.8
4. 3.8 cm, 71
5. 1, 15; 15.03
19.
17.5 N; 70.2
6. below east
7.
x 1, y t3, 7;
x 1 3t,
y 7t
19.
56.5 N; 43.5
6. above east
7, 21
3, 151, 1
10. 7, 10, 7; 14.07
11. 9i 5j 2k
x 3, y 2 t2, 6; x 3 2t,
y 2 6t
5. 4, 1; 4.12
20. x t, y 4t
8. 28, 16, 15
9.
5.0 N; 36.0
114.1 lb
3. 6.0 cm, 219
4. 2.2 cm, 43
Page 352
20. x t, y t 3
7.
21. x t, y 2t 1
3, 11
8. 1, 2, 1
x t,
21. y 12t 54
9.
22. y 2 x 8
10.
1, 10, 7
22.
y 2x
23.
y 21 x 5
4, 3, 4; 6.40
23. y 2 x 8
11. 4 i 2 j 5k
24. about 6.19 ft
24. about 0.68 ft
12.
12; no
13.
6; no
14. 18, 48, 28
15. 10, 24, 47
© Glencoe/McGraw-Hill
25.
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),
H(0, 0, 2)
Bonus:
x t,
y 4t 6
A13
12.
0; yes
13.
0; yes
14.
5, 13, 1
15.
2, 4, 4
25.
A(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),
H(1, 1, 2)
Bonus: x t, y 3t
Advanced Mathematical Concepts
Chapter 8 Answer Key
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,
with Minor
Flaws
• Shows understanding of the concepts vector addition,
subtraction, cross multiplication, dot product, and
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
Satisfactory,
with Serious
Flaws
• Shows understanding of most of the concepts vector
addition, subtraction, cross multiplication, dot product,
and parametric equations.
• 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.
© Glencoe/McGraw-Hill
A14
Advanced Mathematical Concepts
Chapter 8 Answer Key
Open-Ended Assessment
Page 353
1g. Sample answer: 3, 7; The vectors
are perpendicular because their dot
product is zero.
a1b1 a2 b2 7 3 (3)7 0
1a.
( b a
b), as shown in the
1b. a
figure below.
1h.
兩
i j
2 1
1 3
兩
k
0 0 i 0 j 5 k
0
1c. Yes. They are the
same diagonal of
a parallelogram.
2a. x 2 3t
y4t
are shown in
b and b a
1d. No. a
the figures below.
2b. Sample answer: b 6, 2, (1, 3)
3a.
1e. Add the first terms of each vector
together, and then add the second
terms together. These terms
represent the horizontal and vertical
components of the resultant vector,
respectively.
3 8, 1 (11), or
c d
11, 10
The magnitude of
is 兹(
苶1
苶1
苶苶
)2苶
苶
(苶1
苶0
苶苶,
)2 or
c d
about 14.9.
1f. Sample answer: 1, 2, 3 3, 3, 0 4, 1, 3; 4, 1, 3 4i j 3k
© Glencoe/McGraw-Hill
A15
t(56)sin 30 12(32)t2 8 0
4t2 7t 2 0
(4t 1)(t 2) 0
t2
The ball hits the ground after 2
seconds.
兹
苶 , or
3
3b. Distance: x (2)(56) 2 about 97 feet
1, 0
4. Sample answer: The vectors a
and b 0, 1 are perpendicular
because their inner product is
a1b1 a2b2 1(0) 0(1) or 0;
5, 5 are
5, 5, and b
a
perpendicular because their inner
product is a1b1 a2b2 5(5) 5(5) 25 25 0.
Advanced Mathematical Concepts
Chapter 8 Answer Key
Mid-Chapter Test
Page 354
1.
7.05 in., 9.71 in.
2.
39.6 cm
3.
6.6 cm; 64
4.
5.6 cm; 260
Quiz C
Page 356
Quiz A
Page 355
1. 12.93 mm, 1.36 mm
1.
11.4 N; 50.7
2.
12.6 m
2.
19.15 N, 16.07 N
3.
5.5 cm; 29
3.
72.7 N, 234.9
x 1, y 3 t2, 4;
4. x 1 2t, y 3 4t
4.
5.9 cm; 187
3i j
5.
7, 5 ; 8.60
7.
about 8.3 mph
6.
15i 5 j
8.
15, 20
7.
225.25 lb
5.
x t, y 6t 2
9.
3, 4
8.
15, 20
6.
x t, y 25t 45
10.
15, 24
9.
7.
y 16 x 136
11.
4, 3, 12; 13
21, 24
10.
6, 10
8.
y 43 x 334
5.
4, 2 ; 4.47
6.
12.
5i 2 j
13.
8, 10, 1
14.
9, 30, 5
1.
15.
52, 2, 4
2.
Quiz B
Page 355
16.
4; no
3.
17.
0; yes
4.
18.
19.
5, 45, 21
14, 16, 10
5, 0, 12 ; 13
Quiz D
Page 356
x 65t cos 35,
1. y 65t sin 35 16t 2; yes
3i 5 j 4k
3, 131 ,
2.
about 2.33 s
3.
A(0, 1, 0), B(0, 3, 0),
C(0, 3, 3), D(0, 1, 3),
E(3, 1, 0), F(3, 3, 0),
G(3, 3, 3), H(3, 1, 3)
4.
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), H(2, 1, 2)
5.
A(3, 3, 3), B(3, 3, 3),
C(3, 3, 6),
D(3, 3, 6),
E(6, 3, 3), F(6, 3, 3),
G(6, 3, 6), H(6, 3, 6)
10
9; no
18, 38, 25 ;
5. both inner products 0
Sample answer:
20. 3, 24, 12
© Glencoe/McGraw-Hill
A16
Advanced Mathematical Concepts
Chapter 8 Answer Key
Page 357
SAT/ACT Practice
Page 358
Cumulative Review
Page 359
3
1.
D
11.
C
1.
2.
C
12.
C
2.
3.
A
13.
A
A(2, 3), B(2, 1),
3. C(0,4)
4.
4.
E
14.
E
5.
A
15.
D
6.
D
16.
C
C
17.
B
8.
B
18.
C
5.
ƒ(x) 兹2
苶x
苶3
6.
7.
{x x
9.
2 or x
8}
3, 1, 4
2
8. {x 0
7.
1
5
3
1 5
0
x
2 or x
3
1}
232 360k, k is an integer;
Sample answers: 592, 128
10. 12.3 square units
11.
3, 8
units to the right
12. 6
9.
D
19.
12
13.
4兹5
苶
2兹
苶
5x
兹
5
苶 y 兹5
苶
5
5
10.
D
20.
14. 兹5
苶; 333
8
15.
© Glencoe/McGraw-Hill
A17
y 17 x 275
Advanced Mathematical Concepts
0;
Unit 2 Answer Key
Unit 2 Review
2兹
苶
13
苶
1. 13
兹1
苶
苶
7
3. 17
2. 15
4. 1
36. tan x tan x cot2 x
29.
sec x csc x
tan x (1 cot2 x)
sec x csc x
5. 25.9
sin
x 1
cos x sin x 2
6. 5.8
8.
1
4
7. 10.9
9.
2兹
苶
5
5
sec x csc x
1 1
cos x sin x 10. 1
sec x csc x
sec x csc x sec x csc x
30.
11. no solution
37.
12. two; B 72 33′,
c 14.1, and
C 42 27′, or
B 107 27′, c 2.8,
and C 7 33′
13. one; a 40.1,
c 28.1, C 42
sin (180 ) tan cos sin 180 cos cos 180 sin tan cos 0(cos ) (1) sin tan cos sin tan cos cos
tan cos sin cos sin
cos tan cos cos 31.
tan cos tan cos 14. 14.1
15. 78.2
16. 90
17. 135
38.
18. 630
19. 105
40. 2 兹3苶
20. 6.3 in.
21. 0
22. 0
23. 1
24. 1
2
25. 2, 26. none,
3
42.
1
7
25
44.
5 兹2
苶1
苶
2
兹
2
苶
39. 2
41.
43.
1
2
冪
莦5莦莦1兹
莦2苶莦1苶莦
0
4兹
2
苶1
苶
45. 25
46. 0, 90, 180 47. 0
32.
48. 120
5
49.
2兹
1
苶3
苶x 3兹
1
苶3
苶y
13
13
2兹
1
苶3
苶 0; 2兹
1
苶3
苶 ; 56
13
13
50.
5兹
2
苶9
苶x 2兹
2
苶9
苶y
29
29
8兹
2
苶9
苶 0; 8兹
2
苶9
苶 ; 22
29
29
27. none, 2
28.
33. 34
© Glencoe/McGraw-Hill
兹2
苶 兹6
苶
4
2兹
2
苶 35. 2兹2
34. 苶
3
A18
Advanced Mathematical Concepts
Unit 2 Answer Key
51.
(continued)
3兹
1
苶0
苶x
兹
1
苶苶0 y 7兹
1
苶0
苶 0;
10
10
10
5. 15.7 in.
24. 96.2 cm
7兹
1
苶0
苶 ; 342
10
6. v: 11.3 cm; h: 14.4 cm
26.
52. 1.1
53. 3.9
7. x t, y 5t 2
54. 1.5
55. 6.0 cm, 89
8. 6
56. 2.3 cm, 1.5 cm
9. none
苶 11. 7.6
10. 2 兹3
57. 3, 5 58. 1, 1
59. 7, 13 60. 1, 7
k
61. 1, 4, 1; u i 4 j 2兹
1
苶3
苶
13. 13
12. 2; no
28. 28.2 N; 10.3 N
15.
, 29. 2, 2
2
4兹
6苶5
苶
30. 65
31. 335.4 ft
32. 6
34. y 13 x 130
64. 22, 0, 10;
u 22i 10k
35.
16. 18, 12
66. 22
17. 20i 3j 19k
67. 6, 10, 2
69. x 4, y 3 t 2, 2; x 4 2t,
y 3 2t
Unit 2 Test
2
2. 81 units
5兹
2
苶6
苶x
兹2
苶
苶y
兹2
苶
苶 0
6
6
26
26
13
4.
2兹
7
苶
7
18. 40.9
19.
20. 270
21. 231; III
22.
tan x csc x
sec x
sin
x 1
cos x sin x
1
cos x
1
cos x
1
cos x
1
1
1
11
23. 7, 22, 2; yes
© Glencoe/McGraw-Hill
cos (90 A) sin A
cos 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
68. x, y 5 t 1, 5;
x t, y 5 5t
3.
33. a 96.2, B 22, C 32
63. 6, 9, 6;
u 6i 9j 6k
1. true
27. 140
3
14. 2 k
2
62. 13, 3, 7;
u 13i 3j 7k
65. 14
4
25. 27
A19
37. 14
38. 83.8 cm2
39. 40
40. yes
Advanced Mathematical Concepts
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