Vectors

1

• Vectors

Vectors represent quantities that have direction.
Vector quantities are drawn with an arrow. Head-
point of the arrow Tail- the bottom part of the
arrow
2
Two vectors are shown
y
x
3
Vector A is 6 units long. It is pointing to the
right.
y
x
4
Vector A and B are identical.
y
x
5

• They are identical
• because
• they are the same length
• and
• pointing in the same direction.

6
These vectors are not identical.
y
x
7

• They are not identical
• because
• they are not pointing
• in the same direction
• even though they have the
• same length.

8
These three vectors are different.
y
x
9
Two vectors with their tails together.
y
x
10
What is the sum of vector A and B?
y
x
11
Put tail of B to head of A
y
x
12
The resultant goes from tail of A to head of B.
y
x
13
A B B A
y
x
14
What is the negative of vector A?
y
x
15
It is a vector of same length but pointing in the
opposite direction.
y
x
16
What is the value of C - D?
y
x
17
C - D C (-D)
y
x
18
C - D C (-D)
y
x
19
C - D C (-D) E?
y
x
20
What is 2 times vector A?
y
x
21
It is a vector twice as long.
y
x
22
What are the components of C?
y
x
23
Complete the Parallelogram
y
x
24
Locate the x and y components.
y
x
25
Locate the x and y components.
y
x
26
Draw a 2nd vector - Add
y
x
27
What are the components of S?
y
x
28
What are the components of S?
y
x
29
What are the components of CS?
y
x
30
What are the components of CS?
y
x
31
Complete Parallelogram of CS?
y
x
32
Find Resultant of CS?
y
x
33
C S R ( w/components of R)
34
C S R with components of R
35
Vector R with its components
36
Resolving A Vector Into Its Components
37
What we will need

• We will need to be given a vector. The vector we
  will use as an example will be

38
Drawing the vector
39o
E
39
What we will need?

• We will need to be given the direction of the two
  components which we will be asked to find.

In this situation, we will find the horizontal
and the vertical components.
40
Drawing the Component Directions
Vertical
39o
Horizontal
41
Next

• We will find the magnitude in each direction. How?

We will construct a parallelogram with the
diagonal being the vector to be resolved into its
components.
42
Completing the Parallelogram
Vertical
39o
Horizontal
43
Continuing

• Next we will draw in the component vectors which
  we have been asked to find.

44
Drawing the Components
Vertical
39o
Horizontal
45
Identifying the Sides
Vertical
hyp
opp
39o
Horizontal
adj
46
What Trig Function will give the Horizontal
Component?
Vertical
hyp
opp
39o
Horizontal
adj
47
Finding The Horizontal Component
48
Finding The Vertical Component
Vertical
hyp
opp
39o
Horizontal
adj
49
Finding The Vertical Component
50
Here Is Another Vector
Block on Frictionless Incline
Gravity Is Acting
22o
51
Identify Component Directions
22o
52
Vectors at Right Angles Means?

• This means that each component vector will be
  independent of the other.

53
Completing the Parallelogram
22o
54
Draw Component Vectors
Component along plane of ramp
22o
Component Perpendicular to ramp
55
All that is left to do is to find the length of
the two components by whatever method you choose.
22o
56
Find the Angle ?
?
22o
The correct answer is 22 degrees?
57
Next Find the Magnitude of the Two Components
Component along plane of ramp
22o
Component Perpendicular to ramp
58
Here Is an Acceleration Vector
You will be asked to find two components that are
not at right angles.
59
With Components Directions Identified
85o
29o
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With Parallelogram Completed
85o
29o
61
With Component Vectors Drawn
85o
29o
62
All that is left to do is to find the length of
the two components.
85o
29o
63
Determine the Angles in the Lower Triangle
A
85o
C
66o
85o
29o
64
Determine Magnitude of a and b
A
85o
b
c
66o
85o
C
29o
a
B
65
A
b
85o
c
66o
85o
29o
C
a
B
66
Vector Addition
67
Vector Addition

• In this problem we will add three displacement
  vectors, first by using a scale drawing and then
  by using a TI-83 calculator program called
  VADD.

68
The Problem

• A police cruiser chasing a speeding motorist
  traveled 60 km S, then 35 km E 45o N, and
  finally 50 km W. (a) What is the total
  displacement of the cruiser?

69
Selecting a Scale
Select a scale that will allow you to draw an
arrow to represent each displacement.

• This scale must be large enough to be easy to
  work with, yet not so large that the drawing will
  exceed the work area.

70
Selecting a Scale

• A metric scale will be most useful if it is
  divisible by ten.

Possible scale 1 cm 10 km
71
Drawing the Vectorsandfinding theresultant
72
1 cm 10 km
N
Drawing Vectors
E
W
Represent the second displacement
35 km E 45o N
by an arrow 3.5 cm long, with its tail connected
to the head of the first vector.
Represent the first displacement 60 km S
orientated in a north-south orientation with the
arrow at the south end.
Represent the third displacement 50 km W
Draw a line from the tail of the first vector to
the head of the last vector.
Place an arrow on the end of the line where the
line meets the head of the third vector.
Resultant
1
N
3
by an arrow 5 cm long in a west direction.
by an arrow 6 cm long
E
S
2
E
W
73
1 cm 10 km
N
Find Resultant Displacement
E
W
Resultant
Measure the length of the line with your ruler
Convert this length to the magnitude of the
resultant.
Measured length is 4.3 cm
The resultant is
43 km
74
1 cm 10 km
N
Find Resultant Direction
E
W
Resultant
Measure the angle that the resultant is from
south.
Then express the direction using proper notation.
Measured angle is
43 km
36o
S 36o W
75

• The displacement of the cruiser is
• 43 km S 36o W

76
Finding Average Speed
The total distance traveled is
60 km 35 km 50 km 145 km
The time of travel is 1.3 hrs
77
Finding Average Velocity

• The time of travel is 1.3 hrs

The displacement is 43 km S 36o W
78
Note...

• The vector quantity, displacement, was divided by
  a scalar quantity, time.
• When dividing a vector quantity by a scalar
  quantity, the answer is also a vector quantity
  with the same direction as the original vector
  quantity.

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Determining Direction

• All vector directions must be specified as
  relative to a reference direction.

Use East as your reference direction.
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What is the direction of the vector below?
450 north of east
450
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520 South of West (SW)
520
(180-52) 1280) South of East
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Determining Direction
Angles measured counter-clockwise will be
identified as positive and angles measured
clockwise will be identified as negative.