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Chapter 3 - VectorsChapter 3 - Vectors
I.I. Definition Definition
II.II. Arithmetic operations involving vectors Arithmetic operations involving vectors
A) Addition and subtraction A) Addition and subtraction - Graphical method- Graphical method - Analytical method - Analytical method Vector components Vector components
B) Multiplication B) Multiplication
I.I. DefinitionDefinition
Vector quantity:Vector quantity: quantity with a magnitude and a direction. It can be quantity with a magnitude and a direction. It can be represented by a vector.represented by a vector. ExamplesExamples:: displacement, velocity, acceleration. displacement, velocity, acceleration.
Same displacementSame displacement
DisplacementDisplacement does not describe the object’s path. does not describe the object’s path.
Scalar quantity:Scalar quantity: quantity with magnitude, no direction.quantity with magnitude, no direction.
ExamplesExamples: temperature, pressure: temperature, pressure
Vector NotationVector Notation
• When handwritten, use an arrow: When handwritten, use an arrow:
• When printed, will be in bold print:When printed, will be in bold print: AA
• When dealing with just the magnitude of a vector When dealing with just the magnitude of a vector in print:in print: |A||A|
• The magnitude of the vector has physical unitsThe magnitude of the vector has physical units
• The magnitude of a vector is always a positive The magnitude of a vector is always a positive numbernumber
A
Coordinate SystemsCoordinate Systems
• Used to describe the position of a point in Used to describe the position of a point in spacespace
• Coordinate system consists ofCoordinate system consists of– a fixed reference point called the origina fixed reference point called the origin– specific axes with scales and labelsspecific axes with scales and labels– instructions on how to label a point relative to instructions on how to label a point relative to
the origin and the axesthe origin and the axes
Cartesian Coordinate SystemCartesian Coordinate System
• Also called Also called rectangular rectangular coordinate systemcoordinate system
• xx- and- and yy- axes - axes intersect at the originintersect at the origin
• Points are labeledPoints are labeled ((xx,,yy))
Polar Coordinate SystemPolar Coordinate System
– Origin and reference Origin and reference line are notedline are noted
– Point is distancePoint is distance rr from the origin in the from the origin in the direction of angledirection of angle ,, ccwccw from reference from reference lineline
– Points are labeled (Points are labeled (rr,,))
(r,(r,θθ))
Polar to Cartesian CoordinatesPolar to Cartesian Coordinates
• Based on Based on forming a right forming a right triangle fromtriangle from r r andand
• xx = = rr cos cos • yy = = rr sin sin
•
Cartesian to Polar CoordinatesCartesian to Polar Coordinates• rr is the hypotenuse andis the hypotenuse and
an anglean angle
must bemust be ccw ccw from from positivepositive xx axis for these axis for these equations to be validequations to be valid
2 2
tany
x
r x y
•
Review of angle reference systemReview of angle reference system
Origin of angle reference systemOrigin of angle reference systemθ1
0º<θ1<90º
90º<90º<θθ22<180º<180º
θθ22
180º<180º<θθ33<270º<270º
θθ33 θθ44
270º<270º<θθ44<360º<360º
9090ºº
180180ºº
270270ºº
00ºº
θθ44=300=300ºº=-60º=-60º
Angle originAngle origin
ExampleExample
• The Cartesian coordinates of The Cartesian coordinates of a point in thea point in the xyxy plane areplane are ((x,yx,y) = (-3.50, -2.50) m) = (-3.50, -2.50) m, , as as shown in the figure. Find the shown in the figure. Find the polar coordinates of this polar coordinates of this point.point.
• Solution:Solution:
ExampleExample
• The Cartesian coordinates of The Cartesian coordinates of a point in thea point in the xyxy plane areplane are ((x,yx,y) = (-3.50, -2.50) m) = (-3.50, -2.50) m, , as as shown in the figure. Find the shown in the figure. Find the polar coordinates of this polar coordinates of this point.point.
• Solution:Solution:
2 2 2 2( 3.50 m) ( 2.50 m) 4.30 mr x y 2.50 m
tan 0.7143.50 m
216
y
x
Two points in a plane have polar coordinates (Two points in a plane have polar coordinates (2.50 m, 2.50 m, 30.0°30.0°) and) and ((3.80 m, 120.0°3.80 m, 120.0°). Determine (a) the ). Determine (a) the Cartesian coordinates of these points and (b) the Cartesian coordinates of these points and (b) the distance between them.distance between them.
Two points in a plane have polar coordinates (Two points in a plane have polar coordinates (2.50 m, 2.50 m, 30.0°30.0°) and () and (3.80 m, 120.0°3.80 m, 120.0°). Determine (a) the ). Determine (a) the Cartesian coordinates of these points and (b) the Cartesian coordinates of these points and (b) the distance between them.distance between them.
(a)(a) cosx r siny r 1 2.50 m cos30.0x 1 2.50 m sin30.0y
1 1, 2.17, 1.25 mx y
2 3.80 m cos120x 2 3.80 m sin120y 2 2, 1.90, 3.29 mx y
(b)(b) 2 2( ) ( ) 16.6 4.16 4.55 md x y
A skater glides along a circular path of radiusA skater glides along a circular path of radius 5.00 m5.00 m. If he . If he coasts around one half of the circle, find (a) the magnitude coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far the person of the displacement vector and (b) how far the person skated. (c) What is the magnitude of the displacement if he skated. (c) What is the magnitude of the displacement if he skates all the way around the circle?skates all the way around the circle?
A skater glides along a circular path of radiusA skater glides along a circular path of radius 5.00 m5.00 m. If he . If he coasts around one half of the circle, find (a) the magnitude coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far the person of the displacement vector and (b) how far the person skated. (c) What is the magnitude of the displacement if he skated. (c) What is the magnitude of the displacement if he skates all the way around the circle?skates all the way around the circle?
C
B A
5.00 m d
ˆ10.0 10.0 m d i
12 5 15.7 m
2s r
0d
(a)(a)
(b)(b)
(c)(c)
Rules:Rules:
)( lawecommutativabba
)()()( laweassociativcbacba
II. Arithmetic operations involving vectorsII. Arithmetic operations involving vectors
- Geometrical methodGeometrical method
a b
bas
Vector addition:Vector addition: bas
a
b
Vector subtraction:Vector subtraction: )( babad
Vector component:Vector component: projection of the vector on an axis.projection of the vector on an axis.
sin
cos
aa
aa
y
x
x
y
yx
a
a
aaa
tan
22 Vector magnitudeVector magnitude
Vector directionVector direction
aofcomponentsScalar
Unit vector:Unit vector: Vector with magnitude 1.Vector with magnitude 1. No dimensions, no units.No dimensions, no units.
axeszyxofdirectionpositiveinvectorsunitkji ,,ˆ,ˆ,ˆ
jaiaa yxˆˆ
Vector componentVector component
- Analytical method:Analytical method: adding vectors by componentsadding vectors by components.
Vector addition:Vector addition:
jbaibabar yyxxˆ)(ˆ)(
A man pushing a mop across a floor causes it to undergo A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude oftwo displacements. The first has a magnitude of 150 cm150 cm and and makes an angle ofmakes an angle of 120°120° with the positive with the positive xx axis. The axis. The resultant displacement has a magnitude ofresultant displacement has a magnitude of 140 cm140 cm and is and is directed at an angle ofdirected at an angle of 35.0°35.0° to the positive to the positive xx axis. Find the axis. Find the magnitude and direction of the second displacement.magnitude and direction of the second displacement.
A man pushing a mop across a floor causes it to undergo two A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude ofdisplacements. The first has a magnitude of 150 cm150 cm and makes an angle and makes an angle ofof 120°120° with the positive with the positive xx axis. The resultant displacement has a axis. The resultant displacement has a magnitude ofmagnitude of 140 cm140 cm and is directed at an angle ofand is directed at an angle of 35.0°35.0° to the positive to the positive xx axis. Find the magnitude and direction of the second displacement. axis. Find the magnitude and direction of the second displacement.
B R A150 cos120 75.0 cm
150sin120 130 cm
140cos35.0 115 cm
140sin35.0 80.3 cm
x
y
x
y
A
A
R
R
2 2
1
ˆ ˆ ˆ ˆ115 75 80.3 130 190 49.7 cm
190 49.7 196 cm
49.7tan 14.7 .
190
B i j i j
B
θ = 3600 – 14.70 = 345.30
As it passes over Grand Bahama Island, the eye of a hurricane As it passes over Grand Bahama Island, the eye of a hurricane is moving in a directionis moving in a direction 60.060.0 north of west with a speed ofnorth of west with a speed of 41.0 41.0 km/hkm/h. Three hours later, the course of the hurricane suddenly . Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows toshifts due north, and its speed slows to 25.0 km/h25.0 km/h. How far . How far from Grand Bahama is the eyefrom Grand Bahama is the eye 4.50 h4.50 h after it passes over the after it passes over the island?island?
As it passes over Grand Bahama Island, the eye of a hurricane is As it passes over Grand Bahama Island, the eye of a hurricane is moving in a directionmoving in a direction 60.060.0 north of west with a speed ofnorth of west with a speed of 41.0 km/h41.0 km/h. . Three hours later, the course of the hurricane suddenly shifts due north, Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows toand its speed slows to 25.0 km/h25.0 km/h. How far from Grand Bahama is the . How far from Grand Bahama is the eyeeye 4.50 h4.50 h after it passes over the island?after it passes over the island?
Northj
Easti
Natkmhhkmd
WofNatkmhhkmd
ˆ
ˆ
5.3750.1/0.25
600.12300.3/0.41
2
01
km km kmˆ ˆ ˆ ˆ41.0 cos60.0 3.00 h 41.0 sin60.0 3.00 h 25.0 1.50 h 61.5 kmh h h
ˆ144 km
i j j i
j
dtotal=
2 261.5 km 144 km 157 km Magnitude =Magnitude =
Vectors & Physics:Vectors & Physics:
-The relationships among vectors do not depend on the location of the origin of The relationships among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes.the coordinate system or on the orientation of the axes.
-- The laws of physics are independent of the choice of coordinate system The laws of physics are independent of the choice of coordinate system.
'
'' 2222yxyx aaaaa
Multiplying vectors:Multiplying vectors:
- - Vector by a scalar:Vector by a scalar:
- - Vector by a vector:Vector by a vector:
Scalar product Scalar product = scalar quantity= scalar quantity
asf
zzyyxx bababaabba cos
(dot product)(dot product)
)0(cos900
)1(cos0
ifba
ifabbaRule:Rule:abba
090cos11
10cos11
jkkjikkiijji
kkjjii
Multiplying vectors:Multiplying vectors:
- - Vector by a vectorVector by a vector
Vector productVector product = vector = vector
sin
ˆ)(ˆ)(ˆ)(
abc
kabbajabbaiabbacba yxyxxzxzzyzy
(cross product)(cross product)
MagnitudeMagnitude
Angle between two vectors:Angle between two vectors:ba
ba
cos
)( baab
Rule:Rule:
)1(sin90
)0(sin00
ifabba
ifba
DirectionDirection right hand rule right hand rule
bacontainingplanetolarperpendicuc
,
1)1) Place Place aa and and b b tail to tail without altering their orientations. tail to tail without altering their orientations.2)2) cc will be along a line perpendicular to the plane that contains will be along a line perpendicular to the plane that contains a a and and bb
where they meet.where they meet.3) Sweep 3) Sweep aa into into bb through the small angle between them. through the small angle between them.
Vector productVector product
Right-handed coordinate systemRight-handed coordinate system
xx
yy
zz
ij
k
Left-handed coordinate systemLeft-handed coordinate system
yy
xx
zz
iijj
kk
00sin11 kkjjii
jkiik
ijkkj
kijji
kkjjii
)(
)(
)(
0
kbabajbabaibaba
jbaibakbajbakbaiba
bbb
aaa
kji
ba
xyyxzxxzyzzy
zxyzxyxzyxzy
zyx
zyx
ˆ)(ˆ)(ˆ)(
ˆˆˆˆˆˆ
ˆˆˆ
42:: IfIf BB is added tois added to C = 3i + 4jC = 3i + 4j, the result is a vector in the , the result is a vector in the positive direction of thepositive direction of the y y axis, with a magnitude equal to axis, with a magnitude equal to that ofthat of CC. What is the magnitude of. What is the magnitude of BB??
2.319ˆˆ3ˆ5)ˆ4ˆ3(
543
ˆ)ˆ4ˆ3(
22
BjiBjjiB
DC
jDDjiBCB
42:: IfIf BB is added tois added to C = 3i + 4jC = 3i + 4j, the result is a vector in the , the result is a vector in the positive direction of thepositive direction of the y y axis, with a magnitude equal to axis, with a magnitude equal to that ofthat of CC. What is the magnitude of. What is the magnitude of BB??
50: A fire ant goes through three displacements along level A fire ant goes through three displacements along level ground: ground: dd11 for for 0.4m SW0.4m SW, , dd22=0.5m E=0.5m E, , dd33=0.6m=0.6m at at 6060ºº North of North of
East. Let the positive x direction be East and the positive y East. Let the positive x direction be East and the positive y direction be North. (a) What are the direction be North. (a) What are the x x and and y y components of components of dd11, ,
dd22 and and dd33? (b) What are the ? (b) What are the xx and the and the yy components, the components, the
magnitude and the direction of the ant’s net displacement? magnitude and the direction of the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far (c) If the ant is to return directly to the starting point, how far and in what direction should it move?and in what direction should it move?
N
E
d3
d2
45º
d1
D
d4
50: A fire ant goes through three displacements along level A fire ant goes through three displacements along level ground: ground: dd11 for for 0.4m SW0.4m SW, , dd2 2 0.5m E0.5m E, , dd33=0.6m=0.6m at at 6060ºº North of North of
East. Let the positive x direction be East and the positive y East. Let the positive x direction be East and the positive y direction be North. (a) What are the direction be North. (a) What are the x x and and y y components of components of dd11, ,
dd22 and and dd33? (b) What are the ? (b) What are the xx and the and the yy components, the components, the
magnitude and the direction of the ant’s net displacement? magnitude and the direction of the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far and (c) If the ant is to return directly to the starting point, how far and in what direction should it move?in what direction should it move?
N
E
d3
d2
45º
d1
D
md
md
d
md
md
md
y
x
y
x
y
x
52.060sin6.0
30.060cos6.0
0
5.0
28.045sin4.0
28.045cos4.0
3
3
2
2
1
1
(a)(a)
d4
50: A fire ant goes through three displacements along level A fire ant goes through three displacements along level ground: ground: dd11 for for 0.4m SW0.4m SW, , dd2 2 0.5m E0.5m E, , dd33=0.6m=0.6m at at 6060ºº North of North of
East. Let the positive x direction be East and the positive y East. Let the positive x direction be East and the positive y direction be North. (a) What are the direction be North. (a) What are the x x and and y y components of components of dd11, ,
dd22 and and dd33? (b) What are the ? (b) What are the xx and the and the yy components, the components, the
magnitude and the direction of the ant’s net displacement? magnitude and the direction of the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far (c) If the ant is to return directly to the starting point, how far and in what direction should it move?and in what direction should it move?
N
E
d3
d2
45º
d1
D
(b)(b)
EastofNorth
mD
mjijijiddD
mjiijiddd
8.2452.0
24.0tan
57.024.052.0
)ˆ24.0ˆ52.0()ˆ52.0ˆ3.0()ˆ28.0ˆ22.0(
)ˆ28.0ˆ22.0(ˆ5.0)ˆ28.0ˆ28.0(
1
22
34
214
d4
(c) Return vector negative of net displacement, D=0.57m, directed 25º South of West
53:
kjid
kjid
kjid
ˆ2ˆ3ˆ4
ˆ3ˆ2ˆ
ˆ6ˆ5ˆ4
3
2
1
?,)(
?)(
?)(
?)(
2121
21
321
ddofplaneinanddtolarperpendicudofComponentd
dalongdofComponentc
zandrbetweenAngleb
dddra
Find:Find:
53:
kjid
kjid
kjid
ˆ2ˆ3ˆ4
ˆ3ˆ2ˆ
ˆ6ˆ5ˆ4
3
2
1
?,)(
?)(
?)(
?)(
2121
21
321
ddofplaneinanddtolarperpendicudofComponentd
dalongdofComponentc
zandrbetweenAngleb
dddra
θ
d1
d2
kjikjikjikjidddra ˆ7ˆ6ˆ9)ˆ2ˆ3ˆ4()ˆ3ˆ2ˆ()ˆ6ˆ5ˆ4()( 321
mr
rkrb
88.12769
12388.12
7cos7cos1ˆ)(
222
1
md
mdd
ddddd
dd
ddddddc
74.3321
2.374.3
12cos
coscos1218104)(
2222
21
2111//1
21
212121
d1//
d1perp
md
mddddd perpperp
77.8654
16.82.377.8)(
2221
221
21
2//11
30:
kjid
kjidIf
ˆˆ2ˆ5
ˆ4ˆ2ˆ3
2
1
?)4()( 2121 dddd
30:
kjid
kjidIf
ˆˆ2ˆ5
ˆ4ˆ2ˆ3
2
1
?)4()( 2121 dddd
04090cos
),(4)(4)4(
),()(
212121
2121
babtolarperpendicua
planeddtolarperpendicubdddd
planeddincontainedadd
y
x
A
B
130º
54:Vectors Vectors A A and and BB lie in an lie in an xyxy plane. plane. A A has a has a magnitude magnitude 8.008.00 and angle and angle 130130ºº; ; BB has components has components
BBxx= -7.72= -7.72, , BByy= -9.20= -9.20. What are . What are
the angles between the the angles between the negative direction of the negative direction of the y y axis axis and (a) the direction of and (a) the direction of AA, , (b) the direction of (b) the direction of A x BA x B, , (c) the direction of (c) the direction of A x (B+3k)A x (B+3k)??
ˆ
y
x
A
B
130º
1405090)( AandybetweenAnglea
90)(),(
ˆ,ˆ)(,)(
xyBAplane
larperpendicuCbecausekjangleCBAyAngleb
54: Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130º; B has º; B has components Bcomponents Bxx= -7.72, = -7.72, BByy= -9.20. What are the angles between the negative direction of = -9.20. What are the angles between the negative direction of
the y axis and (a) the direction of A, (b) the direction of AxB, (c) the direction of Ax(B+3k)?the y axis and (a) the direction of A, (b) the direction of AxB, (c) the direction of Ax(B+3k)?ˆ
kji
kji
EAD
kjikBE
jijiA
DkBADirectionc
ˆ61.94ˆ42.15ˆ39.18
320.972.7
013.614.5
ˆˆˆ
ˆ3ˆ2.9ˆ72.7ˆ3
ˆ13.6ˆ14.5ˆ)130sin8(ˆ)130cos8(
)ˆ3()(
00
9961.97
42.15
1
ˆcos
42.15)ˆ61.94ˆ42.15ˆ39.18(ˆˆ
61.9761.9442.1539.18 222
D
Dj
kjijDj
D
39: A wheel with a radius ofA wheel with a radius of 45 cm rolls without sleeping along rolls without sleeping along a horizontal floor. At timea horizontal floor. At time t1 the dotthe dot P painted on the rim of the painted on the rim of the
wheel is at the point of contact between the wheel and the wheel is at the point of contact between the wheel and the floor. At a later timefloor. At a later time t2, the wheel has rolled through one-half of , the wheel has rolled through one-half of
a revolution. What are (a) the magnitude and (b) the angle a revolution. What are (a) the magnitude and (b) the angle (relative to the floor) of the displacement(relative to the floor) of the displacement P during this interval?during this interval?
39: A wheel with a radius of 45 cm rolls without sleeping along a horizontal A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time tfloor. At time t1 1 the dot P painted on the rim of the wheel is at the point of the dot P painted on the rim of the wheel is at the point of
contact between the wheel and the floor. At a later time tcontact between the wheel and the floor. At a later time t22, the wheel has , the wheel has
rolled through one-half of a revolution. rolled through one-half of a revolution. What are (a) the magnitude and What are (a) the magnitude and (b) the angle (relative to the floor) of (b) the angle (relative to the floor) of the displacement P during this interval?the displacement P during this interval?
y
xVertical displacement:Vertical displacement:
Horizontal displacement:
d
mR 9.02
mR 41.1)2(2
1
5.322
tan
68.19.041.1
ˆ)9.0(ˆ)41.1(
22
R
R
md
jmimd
Vector Vector a a has a magnitude ofhas a magnitude of 5.0 m5.0 m and is directed East. and is directed East. VectorVector bb has a magnitude ofhas a magnitude of 4.0 m4.0 m and is directedand is directed 3535ºº West West of North. What are (a) the magnitude and direction of of North. What are (a) the magnitude and direction of (a+b)(a+b)? ? (b) What are the magnitude and direction of (b) What are the magnitude and direction of (b-a)(b-a)? (c) Draw a ? (c) Draw a vector diagram for each combination.vector diagram for each combination.
Vector Vector a a has a magnitude ofhas a magnitude of 5.0 m5.0 m and is directed East. and is directed East. VectorVector bb has a magnitude ofhas a magnitude of 4.0 m4.0 m and is directedand is directed 35º35º West West of North. What are (a) the magnitude and direction of of North. What are (a) the magnitude and direction of (a+b)(a+b)? ? (b) What are the magnitude and direction of (b) What are the magnitude and direction of (b-a)(b-a)? (c) Draw a ? (c) Draw a vector diagram for each combination.vector diagram for each combination.
N
Ea
b125º
S
W
jijib
ia
ˆ28.3ˆ29.2ˆ35cos4ˆ35sin4
ˆ5
43.5071.2
28.3tan
25.428.371.2
ˆ28.3ˆ71.2)(
22
mba
jibaa
WestofNorth
or
mab
jiababb
2.248.155180
8.155)2.24(180
2.2429.7
28.3tan
828.329.7
ˆ28.3ˆ29.7)()(
22
a+b
-a
b-a