Name
Date
Class
Reading Strategies
LESSON
8-6
Use a Concept Map
Use the concept map to help you understand vectors.
Component Form
Magnitude
Direction
Lists the horizontal and
vertical change from
the initial point to the
terminal point
The ___length of a vector, written
u
v
as AB or u
The direction of a vector is
measured in degrees.
Initial point P (2, 5)
d
Y
Use the distance formula to
find the magnitude of a vector.
!(5, 6)
2
2
(x 2 x 1)
(y2 y1)
6
2
Terminal point Q (8, 4)
____u
PQ 8 2, 4 5
____u
PQ 6, 1
0
X
5
"
#
2
6
tan B __
5
6 50
m⬔B tan1 __
5
Vectors
Equal Vectors
Parallel Vectors
Adding Vectors
Same magnitude and
same direction
Same direction or opposite
directions
To add vectors numerically,
add their components.
Y
Y
u x 1, y1 and
If u
u
v x 2, y2 , then
2
2
X
0
0
2
u
u u
v x 1 x 2 , y1 y2 .
X
2
Complete the following.
1.
Equal
vectors have the same magnitude and the same direction.
2. Write a vector with initial point (0, 0) and
terminal point (3, 8) in component form.
3, 8
69
3. Find the direction of the vector in Exercise 2 to the nearest degree.
In Exercises 4–6, find the magnitude of each vector. If necessary,
round to the nearest tenth.
4. 3, 4
5
6.3
5. 2, 6
5.1
3, 1
7. Add the vectors in Exercises 5 and 6.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
6. 1, 5
50
Holt Geometry
Name
Date
Class
Name
Reteach
LESSON
8-6
Vectors
Draw the vector �5, 2� on a coordinate plane. Find its magnitude
and direction.
8-6
������
�
�
������
� �
u
3. u
6. ��4, 6�
Draw each vector on a coordinate plane. Find the direction of each vector to
the nearest degree.
8. �6, 3�
������
�
�
�
9. � u
r �
27�
45�
12. cos �
Identify each of the following.
�20
w
�u
5. u
v
0
w
�u
�
������
�
�
�
�
�
�
�
�������
�7, �3�
7. u
s
8. u
r
�
�58 , or 7.6
5
10. � u
s�
0.5
13. m��
19
s
�u
�
5 � 58 , or
11. � u
r �� u
s�
38.1
60�
45�
15. u
p � ��1, �2�, u
q � �4, 2� 143�
u
u
u
u
16. The dot product of vectors j and k is 0. What is the relationship between j and k ?
Explain.
u
c � ��3, 1�, d � ��6, �3�
14. u
�
9. equal vectors
�
u
u
a and d
They are perpendicular. If the dot product is 0, then the numerator of the
�
10. parallel vectors
expression
�
u
b and u
c
u u
r � s
_______
�u
r � �u
s�
equals 0, and the value of the entire expression is 0.
�1
A calculator tells us that cos 0 � 90�.
47
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
8-6
4. u
u
Find the angle between each pair of vectors.
Equal vectors have the same magnitude and the same direction. Parallel vectors
have the same direction or have opposite directions.
LESSON
10
�4, 3�
r
6. u
�
�
�
v
�u
Referring to the figure at right, represent each expression below numerically.
Round to the nearest tenth or whole degree.
�
�
�
�
�
������
��4, 7�
�
The dot product provides a method for calculating the angle
between two vectors. For convenience, consider two vectors,
u
r and u
s , in standard position, that is, with their initial points
at the origin. Let � be the angle formed by u
r and u
s . Then the
Law of Cosines can be used to prove that
u
u
u
�1 u
r
s
r
s
______
______
cos � � u u . So � � cos u u .
� r � �s �
� r � �s �
7.2
7. �4, 4�
�������
u � ��8, �4�. Find each dot product.
u � �1, 3�, u
v � ��2, 4�, and w
Let u
Find the magnitude of each vector to the nearest tenth.
3.2
�
�
2. J (6, �8), K (2, �1)
�
�
�
�
�
�
The dot product of two vectors u
u � �a1, b1� and u
v � �a2, b2� is denoted u
u u
v
and is defined by the rule u
u u
v � a1a2 � b1b2 . For example, if u
u � ��3, 7� and
u
u
u
v � ��2, �1�, then the dot product u v is (�3)(�2) � 7(�1) � �1. Notice that
the dot product is a real number. It is not another vector.
�
�
�
�
�4, 1�
1. J (5, 2), K (9, 3)
�
�
To find the direction, draw right triangle ABC. Then find the measure
of �A.
tan A � _2_
5
�1
m�A � tan _2_ � 22�
5
�
�������
Find the component form of a vector with initial point J and terminal point K.
�
Use the Distance Formula to find the magnitude.
�
��
� �5, 2�� � � (5 � 0)2 � (2 � 0)2 � �29 � 5.4
5. �3, �1�
Vectors on a Coordinate Plane
When a vector is on a coordinate plane, it can be represented by
an ordered pair. If its tail, or initial point, has coordinates (x1, y1)
and its head, or terminal point, has coordinates (x2, y2 ), then its
component form is �x 2 � x 1, y2 � y1 �. For example, the component
u on the coordinate plane at right is
form for vector m
��4 � 5, 2 � (�3)�, or ��9, 5�.
�
To draw the vector, use the origin as the initial point. Then (5, 2) is
the terminal point.
Class
Challenge
LESSON
continued
___u
___u
The magnitude of a vector is its length. The magnitude of AB is written � AB �.
The direction of a vector is the angle that it makes with a horizontal line, such
as the x-axis.
�
Date
Date
Holt Geometry
Class
Problem Solving
8-6
2. Hikers set out on a course given by the
vector �6, 11�. What is the length of the trip
to the nearest unit?
Use the following information for Exercises 3 –5.
A sailboat is traveling in water with a current
shown in the table.
sailboat
3. What is the resultant vector in component
form? Round to the nearest tenth.
current
Direction
Rate
due east
4 mi/h
N 60° E
Component Form
Magnitude
Direction
Lists the horizontal and
vertical change from
the initial point to the
terminal point
The ___length of a vector, written
u
v�
as �AB � or �u
The direction of a vector is
measured in degrees.
6. What is the plane’s actual speed to the
nearest mile per hour?
8. Find the direction of the resultant vector
when you add the given vectors. Round to
the nearest degree.
u
u � ��4, 3� and u
v � �1, 3�
C N 27° W
D N 27° E
0
Rate
plane
due north
200 mi/h
wind
due east
28 mi/h
Parallel Vectors
Adding Vectors
Same magnitude and
same direction
Same direction or opposite
directions
To add vectors numerically,
add their components.
�
�
u
v � �x 2, y2 �, then
2
H 3.2 mi/h
G 0.8 mi/h
J 3.5 mi/h
0
2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
u
u �u
v � � x 1 � x 2 , y1 � y2 �.
�
�
0
2
Complete the following.
1.
9. A person in a canoe leaves shore at a
bearing of N 45° W and paddles at a
constant speed of 2 mi/h. There is a
1.5 mi/h current moving due west.
What is the canoe’s actual speed?
F 0.5 mi/h
u � �x 1, y1 � and
If u
Equal
vectors have the same magnitude and the same direction.
2. Write a vector with initial point (0, 0) and
terminal point (3, 8) in component form.
� 3, 8�
69�
3. Find the direction of the vector in Exercise 2 to the nearest degree.
In Exercises 4–6, find the magnitude of each vector. If necessary,
round to the nearest tenth.
5
6.3
5. � �2, 6�
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
62
6. � �1, �5�
5.1
��3, 1�
7. Add the vectors in Exercises 5 and 6.
49
�
2
Equal Vectors
4. � 3, 4�
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
5
Vectors
7. What is the direction of the plane to the
nearest degree?
F 82°
H 16°
G 41°
J 8°
C 202 mi/h
D 228 mi/h
6
2
� �
2
B N 63° W
� (y2 � y1)
�
6° or N 84° E
A small plane is flying with the conditions shown
in the table.
A N 63° E
�(x 2 � x 1)
tan B � _6_
5
m�B � tan�1 _6_ � 50�
5
Choose the best answer.Use the following information
for Exercises 6 and 7.
A 172 mi/h
B 198 mi/h
d�
�(5, 6)
�
2
2
Terminal point Q (8, 4)
____u
PQ � � 8 � 2, 4 � 5�
____u
PQ � � 6, �1�
1 mi/h
Direction
�
Use the distance formula to
find the magnitude of a vector.
5. What is the sailboat’s actual direction?
Round to the nearest degree.
4.9 mi/h
Holt Geometry
Use a Concept Map
Initial point P (2, 5)
�4.9, 0.5�
4. What is the sailboat’s actual speed to the
nearest tenth?
Class
Use the concept map to help you understand vectors.
13 units
23°
Date
Reading Strategies
LESSON
Vectors
1. The velocity of a wave is given by the
vector �7, 3�. Find the direction of the
vector to the nearest degree.
48
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
50
Holt Geometry
Holt Geometry