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