Read Chapter 1619 Scalars and Vectors All measurements are considered to be quantities In physics there are 2 types of quantities SCALARS AND VECTORS Scalar quantities have only magnitude ID: 757165
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Slide1
Vectors in Two Dimensions
Read Chapter 1.6-1.9Slide2
Scalars and Vectors
All measurements are considered to be quantities. In physics, there are 2 types of quantities –
SCALARS
AND VECTORS.Scalar quantities have only magnitude.Vector quantities have magnitude and direction.
time
mass
temperature
displacement
velocity
acceleration
Gravitational Field
Magnetic Field
ForceSlide3
Vectors are used to describe motion and solve problems concerning motion.
For this reason, it is critical that you have an understanding of how to
represent vectors
add vectors subtract vectors manipulate vector quantities.in 2 and 3 dimensions.Slide4
Vectors
tail
tip
Magnitude
represented by the length of the vector
8 units
5 units
2 unitsSlide5
Vectors
Direction
represented by the direction of the arrow
x
y
0
0
30
0
80
0
120
0
60
0
from -x
225
0
45
0
from-y
300
0
-60
0
Need 3 things to give direction in 2-D
Angle
Axis angle is from
Direction from axisSlide6
Adding Vectors
We know how to add vectors in 1-dimension.
Example:
displacement
. If someone walks 4 mi east (Dx1) and then 7 mi west (Dx2) the total displacement (Dx1
+ Dx
2) is 3 mi west.
1. Adding
vectors GRAPHICALLY
– place them TAIL TO TIP
2. Adding
vectors
MATHEMATICALLY
– In 1- dimension, assign direction + or – and add algebraically
Dx1
+ Dx2
= +4 mi
+ (-7 mi) =
-3 mi
east
+
D
x
1
= 4 mi E
D
x
2
= 7 mi W
D
x
1
+
D
x
2
= 3 mi west
east
west
-3 -2 -1 0 1 2 3 4Slide7
Adding Vectors
What about if the vectors are in different
directions in 2-D?
How do we describe the direction?
How do we add/subtract the vectors?v1 = 5
v
2
= 3
50
o
above +x
35
o
below +xSlide8
Adding Vectors
Graphically
“
tail to
tip”To add vectors using the tail to tip methodDraw the first vector (7u, 50o N of E) beginning at the origin.Draw the second vector (
3u, 35o S
of E) with its tail at the tip of the first vector.
Draw the
Resultant vector (the answer) from the tail of the first vector to the tip of the last.
North
South
East
West
v
1
v
2
v
=
v
1
+
v
2Slide9
Adding Vectors
Grahically
“
parallelogram method
”To add vectors with the parallelogram methodDraw the first vector to scale beginning at the origin.Draw the second vector, to scale, with its tail also at the origin.
Starting at the tip of one vector, draw a dotted line parallel to the other vector. Repeat, starting from the tip of the second vector.
Draw the
Resultant vector (the answer) from the origin to
the intersection of the dotted lines.
North
South
East
West
v
1
v
2
v
=
v
1
+
v
2Slide10
Adding Vectors Mathematically
What
if
the vectors are in different directions?
For example, what if I walk 5 steps north and then 4 steps east. What is my total displacement , Dx, for the trip?OR what is the vector sum of the Dx1 and Dx
2?
D
x
2
=4
steps east
D
x
1
=
5 steps north
D
x
= D
x1+
D
x
2
= ?
North
South
East
WestSlide11
Adding Vectors Mathematically
4 steps east
5 steps north
D
x = ?Use Pythagorean Theorem to find the magnitude of
Dx
Use right triangle trig to find the
direction
of
D
x
qSlide12
RIGHT TRIANGLE TRIGONOMETRY
A
O
H
qA – side adjacent to angle q
O – side opposite to angle qH
– hypotenuse of triangle
sin
q
=
OH
cos
q =
AH
tan
q
=
sin
qcosq
=
O
A
Pythagoreon Theorem
SOHCAHTOASlide13
A student walks a distance of 240 m
East,
then walks
150m south
in 30 min. What is the net displacement? What is the average velocity for the trip?tanq = oppadj150m
240m
=
tan
q =
0.625= tan
-1(1.3) = 32o
south of east
MAGNITUDE
R
2 = A2
+B2
= 2402
+ 1502
R =
283 m
DIRECTION
West
North
South
East
A = 240
m
B=
150m
q
R=283
m
Notice that A and B are perpendicular components of R. They are the amount of R in each directionSlide14
A student walks a distance of 240 m
East,
then walks
150m south
in 30 min. What is the net displacement? What is the average velocity for the trip?West
North
South
East
240
m
150m
q
283
m
The net displacement is
283
m in the direction of
32
o
S
of
E.
The average velocity:
283
m
0.5 hr
=
=
566
m/hr
in the direction
32
o
S
of
E
.
v
av
=
D
x
tSlide15
EXAMPLE:
What is the Resultant of adding 2 vectors, A and B, if A = 8 units south and B = 4.5 u west?
North
South
East
West
4.5u west
8u south
q
tan
q
=
opp
adj
4.5 u
8 u
=
tan
q
= 0.56
q
-1
= tan
-1
(0.56) =
29
o
9.2 u
The Resultant is 9.2 units in the direction of 29
o
south of west
MAGNITUDE
R
2
=
A
2
+
B
2
= 8
2
+ 4.5
2
R
=
9.2 u
DIRECTION
R
Notice that A and B are perpendicular components of R. They are the amount of R in each directionSlide16
30
o
Ex.
Vector
A has magnitude 8.0 m at an angle of 30 degrees below the x-axis. What are the x- and y-components of A?
A=8
A
y
A
x
sin
30
=
A
y
A
A
y
=
8sin
30
= 8(0.5) =
-
4m
cos
30
=
A
x
A
A
x
=
8cos
30
= 8(0.866) =
+6.9
Perpendicular Components of a Vector
Any vector can be resolved into perpendicular components. Use
right triangle trig
– x and y components always make a right triangle with the vector .
A
x
=
A
cos
30
A
y
=
A
sin
30
You must assign the correct directionSlide17
What are the x- and y- components of the vector
A
, shown below?
A =
10 m30o
x
y
A
x
A
y
A
x =
Acosq
= 10cos(30)
= 10(0.866) =
-8.7 m
A
y = Asinq
=
10sin(30
)
=
10(0.5
)
=
+5
m
x- and y- Components of a Vector
A
x
is the amount of A in the x-direction
A
y
is the amount of A in the y-directionSlide18
North
South
East
West
q ?
A butterfly
heads northeast
with a speed of 12
m/s. Its speed in the east direction is 8.00
m/s.
What is the exact direction that the butterfly travels?
v
= 12 m/s
v
y
v
E
= 8
cos
q
=
v
E
v
q
= cos
-1
0.667 = 48
o
8
12
=
N of ESlide19
Adding Vectors by Components
What about adding 2 vectors,
A
and
B
, that are NOT perpendicular or parallel?
y
x
R
y
60
o
A
B
B
y
B
x
R
R
x
Any vector can be described as the
sum
of
perpendicular components
.
Components in the same direction are added
as 1-D vectors to
find the components of the resultant vector.
R
x
=
A
x
+
B
x
=
0
+
B
x
R
y
=
A
y
+
B
y
A
+
B
=
R
(
A
x
+A
y
)
+
(
B
x
+B
y
)
=
(
R
x
+R
y
) Slide20
Adding x- and y- Components of a Vector
R
y
60
o
A
B
B
y
B
x
R
R
x
R
2
=
R
x
2
+
R
y
2
x
Y
A
A
x
A
y
B
B
x
B
y
R
R
x
R
y
A
+
B
=
R
Determine the perpendicular components of each vector. Make a
table
to add up x and y components separately:
q
Magnitude of
R
:
Direction of
R
:
(
A
x
+A
y
)
+
(
B
x
+B
y
)
=
(
R
x
+R
y
)
y
xSlide21
ADDING VECTORS GRAPHICALLY
You
can add as many vectors as you want
North
South
West
B
A
R
C
D
East
A
+
B
+
C
+
D
=
R
Graphically
, the vectors are added “tail to tip”
and the order doesn’t matter
C
+
A
+
D
+
B
=
R
Slide22
A
B
C
D
A
+
B
+ C
+
D =
R
To Add Vectors
by Components:
Place each vector at origin
. Find the x- and y- components of each vector in the sum, and list them in a
table. Make sure to include the direction of each component
by hand.
Add
all the x-components to
find
R
x
.
Add all the y-components
to find
R
y
.
Use Pythagorean Theorem and trigonometry to find the magnitude and direction of
R
.
Mathematically
ADDING VECTORS
(
A
x
+A
y
)
+
(
B
x
+B
y
)
+
(
C
x
+C
y
)
+
(
D
x
+D
y
)
=
(
R
x
+R
y
)
x
Y
A
-
A
x
-
A
y
B
+
B
x
-
B
y
C
+
C
x
+
C
y
D
-
D
x
+
D
y
R
R
x
R
y
R
2
=
R
x
2
+
R
y
2
Magnitude of
R
:
Direction of
R
:Slide23
South
EXAMPLE (Adding Vectors by Components)
Determine the resultant of the following 3 displacements:
A. 24m, 30º north of east B. 28m, 37º east of north C. 20m, 50º west of south x (m)y (m)A
20.812.0
B
16.922.4
C-
15.3 -12.9
Ʃ
22.4
21.5
North
South
West
A
B
C
East
30
o
37
o
50
o
A
y
A
x
B
y
B
x
C
y
C
x
Slide24
South
EXAMPLE
x
(m)
y (m)A20.812.0 B 16.922.4 C-15.3
-12.9
Ʃ R
22.4
21.5
North
South
West
R
East
R
y
R
x
Magnitude
Direction
q
R = 31 m, 44
0
N of E (from the x-axis)Slide25
DO NOWAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-45
0
) for 440 km, and the 3
rd
leg is at 530, south of west, for 550 km. What is the plane’s total displacement?NorthSouthWest
A
B
C
East
45
o
53
o
B
y
C
y
C
x
B
x
x
(km
)
y
(km
)
A
620
0
B
311
-311
C
-
331
-
439
D
R
600
-750
D
R
Slide26
EXAMPLE (cont.)
North
South
West
DR
East
q
x
(km
)
y
(km)
A620
0 B
311
-311 C
-331
-439
D
R
600
-750
Magnitude
Direction
D
R = 960 km, -51
0
from the x-axis (51
0
S of E)Slide27
EXAMPLEAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-45
0
) for 440 km, and the 3
rd
leg is at 530, south of west, for 550 km. What is the plane’s total displacement?NorthSouthWest
D
R
East
q
960 km at 51
0
South of EastSlide28
Subtracting Vectors
In order to subtract a vector, we add the negative of that vector.
The negative of a vector is defined as a vector in the OPPOSITE direction (with each component the negative of the original)
v
1
v
2
-
v
1
-
v
2
+
=
v
1
-
v
2
=
v
1
v
2
-Slide29
A=8
B
=10
60
o
A
+
B = S
A
–
B = D
A
B
S
A
B
D
D
-B
A
Graphical Representation
Tail to tip
Tail to tip
(A and –B)
Tail to tail
(A and
B)
-B
60
oSlide30
A=8
B
=10
60
o
A
+
B = S
A
+ (
–B)
= D
Mathematical Representation
x
y A
+ 80
B
+ 5
- 8.7
R
13
-8.7
x
y
A
+
8
0
-B
-
5
+
8.7
D
3
+8.7
R
R
x
R
y
q
D
D
x
D
y
q
-B
60
oSlide31
Scalar Multiplication
Multiplication of a vector by a
positive
scalar
changes the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar "scales" the vector.Multiplication of a vector by a negative scalar changes the magnitude of the vector, and makes the direction opposite.Example:3A = 3 x
=
=
If the scalar is negative, it changes the direction of the vector.
-
3
A
=
-
3 x
=
=