vector 2016
TRANSCRIPT
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Vectors and Scalars
A vector has magnitude as
well as direction.
Some vector quantities:
displacement, velocity, force,
momentumA scalar has only a magnitude.
Some scalar quantities: mass,
time, temperature
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Distance: A Scalar Quantity
A scalar quantity:
Contains magnitudeonly and consists of anumber and a unit.
(20 m, 40 mi/h, 10 gal)
A
B
Distance is the length of the actual pathtaken by an object.
s = 20 m
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Displacement— A Vector Quantity
A vector quantity:
Contains magnitude
AND direction, anumber, unit & angle.
(12 m, 300; 8 km/h, N)
A
BD = 12 m, 20o
• Displacement is the straight-lineseparation of two points in a specifieddirection.
q
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More about Vectors
• A vector is represented on paper by an
arrow
1. the length represents magnitude
2. the arrow faces the direction of
motion
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A vector is a quantity that has both magnitude and
direction. It is represented by an arrow. The length of
the vector represents the magnitude and the arrow
indicates the direction of the vector.
Two vectors are equal if they have the same direction andmagnitude (length).
Blue and orange
vectors have
same magnitudebut different
direction.
Blue and green
vectors have
same directionbut different
magnitude.
Blue and purple
vectors have
same magnitudeand direction so
they are equal.
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6
Magnitude of a Vector
• The magnitude of a vector is a positive number (with units!)
that describes its size.
• Example: magnitude of a displacement vector is its length.
• The magnitude of a velocity vector is often called speed.
• The magnitude of a vector is expressed using the same letter as
the vector but without the arrow on top of it.
A A Aof Magnitude )(
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P
Q
Initial
Point
TerminalPoint
22, y x
11, y x
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8
Some Vector Properties
• Two vectors that have thesame direction are said to beparallel.
• Two vectors that have
opposite directions are saidto be anti-parallel.
• Two vectors that have thesame length and the samedirection are said to beequal no matter where theyare located.
• The negative of a vector is avector with the samemagnitude (size) butopposite direction
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3-2 Addition of Vectors—Graphical Methods
For vectors in one
dimension, simple
addition and subtraction
are all that is needed. You do need to be careful
about the signs, as the
figure indicates.
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Easy Adding…
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All these planes have the same reading on
their speedometer. (plane speed not speed
with respect to the ground (actual speed)
What
factor is
affecting
theirvelocity?
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A BC
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Addition of Vectors—Graphical Methods
If the motion is in two dimensions, the situation is
somewhat more complicated.
Here, the actual travel paths are at right angles to
one another; we can find the displacement by
using the Pythagorean Theorem.
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Perpendicular VectorsWhen 2 vectors are perpendicular , you may use the
Pythagorean theorem.
95 km,E
55 km, N
A man walks 95 km, East
then 55 km, north.
Calculate his
RESULTANT
DISPLACEMENT.
kmcc
bacbac
8.109120505595Resultant
22
22222
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ExampleA bear, searching for food wanders 35 meters east then 20 meters north.
Frustrated, he wanders another 12 meters west then 6 meters south. Calculate
the bear's displacement.
3.31)6087.0(
6087.23
14
93.262314
1
22
Tan
Tan
m R
q
q
35 m, E
20 m, N
12 m, W
6 m, S
- =23 m, E
- =14 m, N
23 m, E
14 m, N
The Final Answer:
R
3.31,93.26
3.3193.26
m
m
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To add vectors, we put the initial point of the second
vector on the terminal point of the first vector. The
resultant vector has an initial point at the initial point
of the first vector and a terminal point at the terminalpoint of the second vector (see below--better shown
than put in words).
v w
Initial point of vv Move w over keeping
the magnitude and
direction the same.
To add vectors, we pu t the in i t ial point of the second
vector on the term inal poin t of the f irst vector . The
resultant vector has an initial point at the initial point
of the first vector and a terminal point at the terminalpoint of the second vector (see below--better shown
than put in words).
To add vectors, we put the initial point of the second
vector on the terminal point of the first vector . The
resultant vecto r has an ini t ia l po int at the ini t ia l po int
of the first vecto r and a term inal point at the term inalpo in t of the second vector (see below--better shown
than put in words).Terminal
point of w
w
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18
Vector Addition• Vector C of a vector sum of vectors A and C.
• Example: double displacement of particle.
• Vector addition is commutative (the order of vector
addition doesn’t matter).
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The negative of a vector is just a vector going the opposite
way.
v
v
A number multiplied in front of a vector is called a scalar . It
means to take the vector and add together that many times.
v
v
vv3
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u
v
wvu u
v
w3w w
w
Using the vectors shown,
find the following:
vuu
vvwu 32
uu w
w w
v
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Vector Subtraction
• Subtract vectors:
)( B A B A
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3 4 Addi V t b C t
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3-4 Adding Vectors by Components
Any vector can be expressed as the sum
of two other vectors, which are called its
components. Usually the other vectors are
chosen so that they are perpendicular to
each other.
3 4 Addi V t b C t
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3-4 Adding Vectors by Components
If the components are
perpendicular , they can be
found using trigonometric
functions.
Remember:
soh
cah
toa
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Adding Vectors by Components
A
B
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Adding Vectors by
Components
A B
Transform vectors so they are head-to-
tail.
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Adding Vectors by
Components
A B B
y
Bx
Ax
Ay
Draw components of each vector...
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Adding Vectors by
Components
A B
By
BxAx
Ay
Add components as collinear vectors!
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Adding Vectors by
Components
A B
Combine components of answer using the head to tail
method...
Ry
Rx
R
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Analytical (component)
Method• Polar Form of Vectors
• = ,
= =
• Example
= 10 , 45° = 10 = 450
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Caution
• Addition of vectors in polar form cannot bedone algebraically
Ex. A = 5 km, 45 deg
B = 4 km, 135 deg
C = 3 km, 270 deg
R = 12 km, 450 deg
Vectors can only be added algebraically ifthey are parallel or antiparallel
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Component Form
• = ,
• = cos + sin
– and – = 10 , 45°
– = 10 (45) + 10 sin 45 – = 7.07 + 7.07
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= 10 , 45°
= 10 (45) + 10 sin 45 = 7.07 + 7.07
7.07
7.07 10
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= 20 , 120°
= 20 (120) + 20 sin 120 = 10 + 17.32
17.32 20
10
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• = 7.07 + 7.07 • = 10 + 17.32
• R = -2.93 km x + 24.39 km y
17.32
10
7.07
7.07
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Graphical Representation of the
Analytical Method• = 12 km, 30 deg
• = 6 km, 60 deg
• + =
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= +
= +
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Component Method
x y
D1 620km, 0 deg 620km (cos 0) 620km (sin 0)D2 440km, 315 deg 440km (cos 315) 440km (sin 315)
D3 550 km, 233 deg 550km (cos 233) 550km (sin 233)
x y
D1 620km, 0 deg 620km 0kmD2 440km, 315 deg 311km -311 km
D3 550 km, 233 deg -331 km -439 km
R 600 km -750 km
• ++ =
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Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines andcosines.
5. Add the components in each direction.
6. To find the magnitude of the vector, use:
= +
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Direction:
1. Resolve what quadrantis the vector pointing at?
2. Get the Reference Angle =−
3. if Quadrant 1 = Quadrant 2 = Quadrant 3 = +
Quadrant 4 =
+x
+y
–x
+y
– x
– y +x – y
Q1Q2
Q3 Q4
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From Component to Polar
• Magnitude of R = 600 + 750 =960
• Angle of R = − − =51 deg at Q4• 360-51 = 309 degrees
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= 600 750
750
600
Adding Vectors by
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Adding Vectors by
Components
Mail carrier’s displacement.
A rural mail carrier leaves the post office and drives
22.0 km in a northerly direction. She then drives in a
direction 60.0
south of east for 47.0 km. What is her
displacement from the post office?
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Magnitude Direction
22 90
47 -60
X Y
0 22
23.5 -40.70319398
23.5
-18.70319398
=
+
= 23.5 + 18.7
= 23.5 + 18.7 =30.0
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Scaling Vectors
x y
3*D1 = 1860 km, 0 deg
2*D2 = 880 km, 315 deg4*D3 = 2200 km, 233 deg
R
3 1 2 2 + 4 3 =
D1 620km, 0 deg 3*D1 = 1860 km, 0 deg
D2 440km, 315 deg 2*D2 = 880 km, 315 degD3 550 km, 233 deg +4*D3 = 2200 km, 233 deg