CONFIDENTIAL1 Geometry Vectors. CONFIDENTIAL2 Warm up Find each measure. Round lengths to the...
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Transcript of CONFIDENTIAL1 Geometry Vectors. CONFIDENTIAL2 Warm up Find each measure. Round lengths to the...
CONFIDENTIAL 1
Geometry
Vectors
CONFIDENTIAL 2
Warm up
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
1) BC 2) m/B 3) m/C
50
3.5
4CA
B
CONFIDENTIAL 3
Vectors
You can think of a vector as a directed line
segment. The vector below line segment. The
vector below may be named AB or v
Initial point
A
Terminalpoint
B
CONFIDENTIAL 4
A vector can also be named using component from. The
Component from 4 {x, y} of a vector lists the horizontal and
vertical change from the initial point to the terminal point. The
Component from of CD is {2,3}.
2
3
C
D
CONFIDENTIAL 5
Writing Vectors in Component From
write each vector in component from.
A) EF The horizontal change from E to F is 4 units. The vertical change from E to F is -3 units. So the comoponent from of EF is {4,-3}.
B) PQ with p(7,-5) and Q(4,3) PQ = (x2 - x1, y2 - y1)
PQ = {4 - 7, 3 - (-5)}
PQ = (-3,8)
Subtract the coordinates of the initial point from the coordinates of the terminal point.
Substitute the coordinate of the given points simplify.
E
F
CONFIDENTIAL 6
Now you try!
u
1) Write each vector in component from.
a) u
b) The vector with initial point L(-1,1) and terminal point M(6,2)
CONFIDENTIAL 7
The magnitude of a vector is its length. The magnitude
is written |AB| or | v |.
When a vector is used to represent speed in a given
direction, the magnitude of the vector equals the speed. For
example, if a vector represents the course a kayaker
paddles, the magnitude of the vector is the kayaker’s speed.
CONFIDENTIAL 8
Finding the magnitude of a Vector
Step 1 Draw the vector
on a coordinate plane.
Use the origin as the
initial point. Then (4,-2)
is the terminal point.
Step 2 Find the magnitude.
Use the Distance Formula.
(4,-2) = (4-0) 2 + (-2-0) 2 = 20 4.5
0x
y
(4,-2)
4-2
2
CONFIDENTIAL 9
Now you try!
2)Draw the vector (-3,1) on a coordinate plane. Fine its
magnitude to the nearest tenth.
CONFIDENTIAL 10
The direction of a vector is the angle that it
makes with a horizontal line. This angle is
measured counterclockwise from the
positive x -axis.
The direction of AB is 60˚.
B
A
60
CONFIDENTIAL 11
The direction of a vector can also be given as a bearing
relative to the compass directions north, south, east, and west . AB has a bearing of N 30˚ E.
B
W E
S
N
30˚
A
CONFIDENTIAL 12
Finding the Direction of a vector
A wind velocity is given by the vector (2,5).
Draw the vector on a coordinate plane. Find the
direction of the vector to the nearest degree.
Cy
4
2
(2,5)
4A B
Step 1 Draw the vector on a coordinate plane.
Use the origin as the initial point.
Next Page ->
CONFIDENTIAL 13
Step 2 Find the direction.
Draw right triangle ABC as shown. /A is the angle formed by the vector and the x –axis, and
Cy
4
2
(2,5)
4A B
tan A = 5
2. So mA = tan-1
5
2 68
CONFIDENTIAL 14
Now you try!
3) The force exerted by a tugboat is given by the
vector (7,3). Draw the vector on a coordinate plane.
Find the direction of the vector to the nearest
degree.
CONFIDENTIAL 15
u
v
Two vectors are equal vector if they have the same
magnitude and the same direction For example, u = v.
Equal vector do not have to have the same initial
point and terminal point.
| u | = | v | = 2√5
Next Page ->
CONFIDENTIAL 16
Two vectors are parallel vectors if they have the same
direction or if they have opposite directions. They may
have different magnitudes. For example, w || x . equal
vectors are always parallel vectors.
x
|w| = 2√5 |x| =√5
w
CONFIDENTIAL 17
A
B
C
D
F
E
G
H
Identify each of the following.
A) AB = GH Identify vectors with the same
magnitude and direction.
B) Parallel vectors
AB || GH and CD || EF Identify vectors with the
same or opposite directions.
Identifying Equal and Parallel Vectors
CONFIDENTIAL 18
M
Y
XS
R
Q
PN
Now you try!
4) Identify each of the following.
a) Equal vectors
b) Parallel vectors
CONFIDENTIAL 19
The resultant vector is the vector that represents the
sum of two given vectors. To add two vectors
geometrically, you can use the head-to-tail method or
the parallelogram method.
CONFIDENTIAL 20
Vector Addition
METHOD EXAMPLE
Head-to-tail method
Place the initial point (tail) of the second vector. On the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector.
u = v
u
v
Next Page ->
CONFIDENTIAL 21
Vector Addition
METHOD EXAMPLE
Parallelogram Method
Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed.
u = v
u
v
CONFIDENTIAL 22
To add vectors numerically, and their components.
If u = (x ,y ) and v = (x ,y ), then
u + v =(x +x , y +y ).21
1 1 2 2
12
CONFIDENTIAL 23
Sports Application
A kayaker leaves shore at a bearing of N 55˚ E and paddles at a constant speed of 3 mi/h. There is 1 mi/h current moving due east. What are the kayak’s actual speed and direction?
Round the speed to the nearest tenth and the direction to the nearest degree.
35˚55˚
3
X
y
N
E E
S S
W W
N
Next Page ->
kayaker Current
1
Step 1 Sketch vectors for the kayaker and the current.
CONFIDENTIAL 24
35˚55˚
3
X
y
N
E E
S S
W W
N
Next Page ->
Step 2 Write the vector for the kayaker in component from. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35˚ with the x –axis.
cos 35 =x
3, so x = 3cos 35 2.5
sin 35 =y
3, so y = 3sin35 1.7.
The kayaker's vector is (2.5, 1.7).
1
kayaker
Current
CONFIDENTIAL 25
Step 3 Write the vector for the current in component from.
Since the current moves 1 mi/h in the direction of the x –axis, it has a horizontal component of 1 and vertical component of 0. So its vector is (1,0).
35˚55˚
3
X
y
N
E E
SS
W
N
W 1
Currentkayaker
Next Page ->
CONFIDENTIAL 26
Step 4 Find and sketch the resultant vector AB.
Add the components of the kayaker’s vector and the current’s vector.
(2.5, 1.7) + (1,0) = (3.5, 1.7)
The resultant vector in the component from is (3.5, 1.7).
S
W
N
resultant
3.5
1.7
(3.5, 1.7)
E
B
C
A
Next Page ->
CONFIDENTIAL 27
Step 5 Find the magnitude and direction of the resultant vector.
The magnitude of the resultant vector is the kayak’s actual speed.
|(3.5,1.7)| = (3.5 - 0)2 + (1.7 - 0 )2 3.9 mi/h
The angle measure formed by the resultant vector gives the kayak’s actual direction.
tan A = 1.7
3.5, so A = tan-1
1.7
3.5 26 , or N 64 E.
S
W
Nresultant
3.5
1.7
(3.5, 1.7)
E
B
CA
CONFIDENTIAL 28
Now you try!
5) What if….? Suppose the kayaker in Example 5 instead
Paddles at 4 mi/h at a bearing of N 20˚ E. What are the
kayak’s actual speed and direction? Round the speed to
the nearest tenth and the direction to the nearest degree.
CONFIDENTIAL 29
Now some practice problems for you!
CONFIDENTIAL 30
Q
P
Write each vector in component from.
1) PQ
2) AC with A(1,2) and C(6,5)
Assessment
CONFIDENTIAL 31
Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.
3) (1,4) 4) (-3,-2) 6) (5, -3)
CONFIDENTIAL 32
Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree.
6) A river’s current is the given by the vector (4,6).
7) The path of a hiker is given by the vector(6,3).
CONFIDENTIAL 33
Diagram 1 Diagram 2
Identify each of the following.
8) Equal vectors in diagram 1
9) Parallel vectors in diagram 2
A
Y
X
S
R
Q
P
N
M
H
G
F
E
D
C
B
CONFIDENTIAL 34
10) To reach a campsite, a hiker first walks for 2 mi at a bearing of N 40˚ E. Then he walks 3 mi due east. What are the magnitude and direction of his hike from his starting point to the campsite? Round the distance to the nearest tenth of a
mile and the direction to the nearest degree.
40˚
campsite
S
W
N
E
2 mi
3 mi
CONFIDENTIAL 35
Let’s review
VectorsYou can think of a vector as a directed line
segment. The vector below line segment. The
vector below may be named AB or v
Initial point
A
Terminalpoint
B
CONFIDENTIAL 36
A vector can also be named using component from. The
Component from 36 {x, y} of a vector lists the horizontal and
vertical change from the initial point to the terminal point. The
Component from of CD is {2,3}.
2
3
C
D
CONFIDENTIAL 37
Writing Vectors in Component From
write each vector in component from.
A) EF The horizontal change from E to F is 4 units. The vertical change from E to F is -3 units. So the comoponent from of EF is {4,-3}.
B) PQ with p(7,-5) and Q(4,3) PQ = (x2 - x1, y2 - y1)
PQ = {4 - 7, 3 - (-5)}
PQ = (-3,8)
Subtract the coordinates of the initial point from the coordinates of the terminal point.
Substitute the coordinate of the given points simplify.
E
F
CONFIDENTIAL 38
The magnitude of a vector is its length. The magnitude
is written |AB| or | v |.
When a vector is used to represent speed in a given
direction, the magnitude of the vector equals the speed. For
example, if a vector represents the course a kayaker
paddles, the magnitude of the vector is the kayaker’s speed.
CONFIDENTIAL 39
Finding the magnitude of a Vector
Step 1 Draw the vector
on a coordinate plane.
Use the origin as the
initial point. Then (4,-2)
is the terminal point.
Step 2 Find the magnitude.
Use the Distance Formula.
(4,-2) = (4-0) 2 + (-2-0) 2 = 20 4.5
0x
y
(4,-2)
4-2
2
CONFIDENTIAL 40
The direction of a vector is the angle that it
makes with a horizontal line. This angle is
measured counterclockwise from the
positive x -axis.
The direction of AB is 60˚.
B
A
60
CONFIDENTIAL 41
The direction of a vector can also be given as a bearing
relative to the compass directions north, south, east, and west . AB has a bearing of N 30˚ E.
B
W E
S
N
30˚
A
CONFIDENTIAL 42
Finding the Direction of a vector
A wind velocity is given by the vector (2,5).
Draw the vector on a coordinate plane. Find the
direction of the vector to the nearest degree.
Cy
4
2
(2,5)
4A B
Step 1 Draw the vector on a coordinate plane.
Use the origin as the initial point.
Next Page ->
CONFIDENTIAL 43
Step 2 Find the direction.
Draw right triangle ABC as shown. /A is the angle formed by the vector and the x –axis, and
Cy
4
2
(2,5)
4A B
tan A = 5
2. So mA = tan-1
5
2 68
CONFIDENTIAL 44
u
v
Two vectors are equal vector if they have the same
magnitude and the same direction For example, u = v.
Equal vector do not have to have the same initial
point and terminal point.
| u | = | v | = 2√5
Next Page ->
CONFIDENTIAL 45
Two vectors are parallel vectors if they have the same
direction or if they have opposite directions. They may
have different magnitudes. For example, w || x . equal
vectors are always parallel vectors.
x
|w| = 2√5 |x| =√5
w
CONFIDENTIAL 46
Identifying Equal and Parallel Vectors
Identify each of the following.
A) AB = GH Identify vectors with the same
magnitude and direction.
B) Parallel vectors
AB || GH and CD || EF Identify vectors with the
same or opposite directions.
A
B
C
D
F
E
G
H
CONFIDENTIAL 47
The resultant vector is the vector that represents the
sum of two given vectors. To add two vectors
geometrically, you can use the head-to-tail method or
the parallelogram method.
CONFIDENTIAL 48
Vector Addition
METHOD EXAMPLE
Head-to-tail method
Place the initial point (tail) of the second vector. On the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector.
u = v
u
v
Next Page ->
CONFIDENTIAL 49
Vector Addition
METHOD EXAMPLE
Parallelogram Method
Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed.
u = v
u
v
CONFIDENTIAL 50
To add vectors numerically, and their components.
If u = (x ,y ) and v = (x ,y ), then
u + v =(x +x , y +y ).21
1 1 2 2
12
CONFIDENTIAL 51
Sports Application
A kayaker leaves shore at a bearing of N 55˚ E and paddles at a constant speed of 3 mi/h. There is 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest
tenth and the direction to the nearest degree.
35˚55˚
3
X
y
N
E E
S S
W W
N
Next Page ->
kayaker Current
1
Step 1 Sketch vectors for the kayaker and the current.
CONFIDENTIAL 52
35˚55˚
3
X
y
N
E E
S S
W W
N
Next Page ->
Step 2 Write the vector for the kayaker in component from. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35˚ with the x –axis.
cos 35 =x
3, so x = 3cos 35 2.5
sin 35 =y
3, so y = 3sin35 1.7.
The kayaker's vector is (2.5, 1.7).
1
kayaker
Current
CONFIDENTIAL 53
Step 3 Write the vector for the current in component from.
Since the current moves 1 mi/h in the direction of the x –axis, it has a horizontal component of 1 and vertical component of 0. So its vector is (1,0).
35˚55˚
3
X
y
N
E E
SS
W
N
W 1
Currentkayaker
Next Page ->
CONFIDENTIAL 54
Step 4 Find and sketch the resultant vector AB.
Add the components of the kayaker’s vector and the current’s vector.
(2.5, 1.7) + (1,0) = (3.5, 1.7)
The resultant vector in the component from is (3.5, 1.7).
S
W
N
resultant
3.5
1.7
(3.5, 1.7)
E
B
C
A
Next Page ->
CONFIDENTIAL 55
Step 5 Find the magnitude and direction of the resultant vector.
The magnitude of the resultant vector is the kayak’s actual speed.
|(3.5,1.7)| = (3.5 - 0)2 + (1.7 - 0 )2 3.9 mi/h
The angle measure formed by the resultant vector gives the kayak’s actual direction.
tan A = 1.7
3.5, so A = tan-1
1.7
3.5 26 , or N 64 E.
S
W
Nresultant
3.5
1.7
(3.5, 1.7)
E
B
CA
CONFIDENTIAL 56
You did a great job today!