7.1 Scalars and vectors Scalar: a quantity specified by its magnitude, for example: temperature,...

7.1 Scalars and vectors

Scalar: a quantity specified by its magnitude,

for example: temperature, time, mass, and density

Chapter 7 Vector algebra

Vector: a quantity specified by a magnitude and a direction,

for example: force, momentum, and electric field

a

Vector

)( baba

7.2 Addition and subtraction of vectors

cbacba

abba

)()( :eAssociativ

:eCommutativ


a

b

a

b

ba

ab

ba

ba

a

b

b

x

aaa

baba

aaa

)(

)(

)()()(


7.3 Multiplication by a scalar

a

a

Ex: A point P divides a line segment AB in the ratio λ: μ. If the position

vectors of the point A and B are and respectively, find the position

vector of the point P.

a

b

a

b

A

B

O

p

P

baba

aba

BAaPAaPO

)1(

)(

Ex: The vertices of triangle ABC have position vector and relative

to some origin O. Find the position vector of the centroid G of the triangle.


ba

, c

a

A

b

c

E

B

F

C

D

G

3/)(

3

2

2

11 ,1

2

1 ,

)(2

1)1()(

2

1)1(

:Gpoint for

)(2

1)1()1(

is BE line onvector point the

)(2

1)1()1(

)-(1 : ratio the in line the dividesthat CD line

the onpoint general a ofvector position the2

1

2

1 ,

2

1

2

1

cbagGO

cabbac

rr

cabebr

bacdcr

caeEObadDO

BECD

BE

CD

d

e


7.4 Basis vector and components

basis the torespect

with vector the of components the are and ,*

basis a form and , , vectors three*

321

321

aaaa

eee

332211 eaeaeaa

A basis set must

(1) have as many basis vectors as the number of dimension

(2) be such that the basis vectors are linearly independent

In Cartesian coordinate system

0...except 0......... 212211 NNN cccececec

),,( zyx

kbajbaibaba

aaakajaiaa

zzyyxx

zyxzyx

ˆ)(ˆ)(ˆ)(

),,(ˆˆˆ

7.4 Basis vectors and components


7.5 Magnitude of a vector

||/ˆvector unit

vector of magnitude the is || 222

aaa

aaaaaa zyx

7.6 Multiplication of vectors (1) scalar product (2) vector product

baba

(1) Scalar product:

abb

abba

of direction the onto of projection the is cos

0 cos

a

b

O cosb

The Cartesisn basis vectors

are mutually orthogonal

ˆ and ˆ ,ˆ kji

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

ikkjji

kkjjii

Ex: work:

potential energy:

rFW

BmE

Commutative and distributive:


cabacba

abba

)(

In terms of the components, the scalar product is given by

)ˆˆˆ()ˆˆˆ(

x zzyyx

zyxzyx

bababa

kbjbibkajaiaba

Ex: Find the angle between the vector

and

kjia ˆ3ˆ2ˆ

kjib ˆ4ˆ3ˆ2

rad 12.09926.02914

20cos

29432 14321

20433221 cos

222222

ba

baab

ba


direction cosines of vector a

a

a

a

ka

a

a

a

ja

a

a

a

ia

zz

yy

xx

ˆcos

ˆcos

ˆcos

scalar product for vectors with complex components

aaaa

baba

baba

abba

babababa zzyyxx

�

||

)(

)(

)(*

*

***

(1) Vector product:

and both tolar perpendicu direction

sin|||| is magnitude

ba

baba


a

b

ba

Properties:

baba

aa

cbacba

baab

cbcacba

to elantiparallor parallel is 0

0

)()(

)(

)(

Ex: The moment or torque about O is

sin|||||| and FrFr

F

r

O

Ex: If a solid body rotates about some axis, the velocity of any point in the body with position vector is .


r

rv

For the basis vector in Cartesian coordinate:

ˆˆˆ

ˆ)(ˆ)(ˆ)(

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

0ˆˆˆˆˆˆ

zyx

zyx

xyyxzxxzyzzy

bbb

aaa

kji

kbabajbabaibababa

jkiik

ijkkj

kijji

kkjjii

Ex: find and the area of

parallelogram.

kjibkjia ˆ6ˆ5ˆ4 ,ˆ3ˆ2ˆ


ba

54)3(6)3(||

ˆ3ˆ6ˆ3

654

321

ˆˆˆ

222

baA

kji

kji

ba

a

b sin|| b

||sin||||2

12 baba

Scalar triple product ],,[)( cbacba

v

a

c

b

O

ipedparallelep a of volume

cos))(sin()(

cos cos

sin

cabcvcba

OPcvccv

abvbav

coplanar are and ,0)( cbacba

)()()()()(

)(

cababcbcaacbbac

aaa

ccc

bbb

bbb

aaa

ccc

ccc

bbb

aaa

cba

zyx

zyx

zyx

zyx

zyx

zyx

zyx

zyx

zyx


Useful formulas:

0)()()( (3)

)()()( (2)

identity sLagrange' ))(())(()()( (1)

bacacbcba

cbabcacba

cbdadbcadcba

Some basic operations:

equal are and,, of two any 0

npermutatio odd 1

npermutatio even 1

:symbol Civita-Levi (2)

if 0

if 1 :deltaKronecker (1)

kji

ε

ji

ji

jminjnimmnkk

ijkijk

ij

Ex: Show that

k

kii

ikii

iiki

iki

ii

nimknki nm

minmiikjmnjnmji nm

i

ijkmnji m n

nmj

iijkji j

i

k

cbabca

cbabcacbacba

cbacba

cbacba

cba

])()[(

)()(

][)(

)(

)]([

,, ,


cbabcacba

)()()(

ijkji j

ik

ii

iiji j

ji

baba

kjibababa

)(

3,2,1 3Dfor

Proof:

Equation of a line:

A line passing through the fixed point A with position vector and having a

direction , the position vector of a general point R on the line is


7.7 Equations of lines, planes and sphere

a

b

r

0)()( bbbarbarbar

a

r

O

b

A

R0)( :equation Line bar

Equation of a plane:


n̂

a r

d

AR

O

equation plane

ˆˆˆˆ and ˆˆˆ

ˆˆ0ˆ)(

plane a ofvector normalunit the :ˆ

vector a by drepresente plane, a onpoint general :R

vector a by drepresente plane, a onpoint fixed :A

dnzmylx

knjmilnkzjyixr

dnanrnar

n

r

a

The equation of a plane containing points A, B and C with position vectors cba

and , ,

1

)ˆ()ˆ()ˆ(ˆ

)ˆ()ˆ()ˆ()ˆ(

)()(

dddd

cnbnanrn

dcnbnanrn

cbar

acabar

a b

O

c

A

B

Cab

ac


Ex: Find the direction of the line of intersection of the two planes

x+3y-z=5 and 2x-y+4z=3.

Normal vector of the planes are kjinkjin ˆ4ˆˆ2 ˆˆ3ˆ21

The direction vector of line is along the direction of

kji

kji

nn ˆ8ˆ6ˆ10

412

131

ˆˆˆ

21

Equation of a sphere with radius a:22 )()(|| acrcrcr

O

r

c

acr


Ex: Find the radius of the circle that is the intersection of the plane and the sphere of radius centered on the point with position vector .

prn ˆc

O

c

b

r

n̂

plane

2222

2222

222222222

222

22

222222

22

22

)ˆ()ˆ(-

ˆ and 2||for

)ˆ(2)ˆ(2

2)()(||

ˆˆ

||

ˆˆ||)(

circle ngintersecti the on vectro position a :

circle the ofcenter the ofvector position the :

sphere a ofcenetr the ofvector position the :

|| :equation circle ngintersecti The

|| :equation sphere The

ncpaancp

prnaccrrcr

aanccnacr-r

bbrrbrbrbr

nacncb

aacb

ncbncb

r

b

c

br

acr

a

7.8 Using vectors to find distances

The minimum distance from a point to a line


O

A

a

P

dp

ap

b�|ˆ)(|sin|| bapapd

Ex: Find the minimum distance from the point P with coordinate (1,2,1) to the line , where bar

kjibkjia ˆ3ˆˆ2 ,ˆˆˆ

14

13)32(

14

1

]ˆ3ˆ2[14

1]ˆ3ˆˆ2[

14

1ˆˆ)(

ˆˆ2ˆ ,)ˆ3ˆˆ2(14

1ˆ

22

d

ikkjijbap

kjipkjib

bb

The minimum distance from a point to a plane

situated. is P plane

the of side which on depends of sign The *

ˆ)(

d

npad


P

a

p

d

n̂

ap

O

Ex: Find the distance from the point P with coordinate (1,2,3) to the plane that contains the point A, B and C having coordinates (0,1,0), (2,3,1) and (5,7,2).

origin the from plane the of side opposite the on is P(1,2,3)3/1 P(0,0,0),for

3/53/)ˆ2ˆˆ2()ˆ3ˆˆ(ˆ)(

3/)ˆ2ˆˆ2(||/ˆ

ˆ2ˆˆ2)()(

ˆ2ˆ6ˆ5 ,ˆˆ2ˆ2

d

kjikjinpad

kjinnn

kjiacabn

kjiackjiab

The minimum distance from a line to a line


|ˆ)(|||

ˆ nqpdba

ban

n̂

a

b

Q

P

q

p

pq

Ex: A line is inclined at equal angles to the x-, y- and z-axis and pass through the origin. Another line passes through the points (1,2,4) and (0,0,1). Find the minimum distance between the two lines.

6/1|6/)ˆˆ2ˆ(ˆ|ˆ

6/)ˆˆ2ˆ(ˆ

ˆˆ2ˆ)ˆ3ˆ2ˆ()ˆˆˆ(

)ˆ3ˆ2ˆ(ˆ ,)ˆˆˆ(0 21

kjikdkqp

kjin

kjikjikjin

kjikrkjir

The distance from a line to a plane


bar

|ˆ)(|0ˆplane a to parallel is line A(2)

0 ,0ˆplane a to parallelnot is line A(1)

nradnb

dnb

n̂

r

O

a

b

ra

Ex: A line is given by , where and Find the coordinates of the point P at which the line intersects the plane x+2y+3z=6.

kjibkjiabar ˆ6ˆ5ˆ4 ˆ3ˆ2ˆ

point ngintersecti the is )2/3,4/3,0(2/3 and 4/3 ,0

-1/4eq. plane intoput 63 and 52 ,14

ˆ)63(ˆ)52(ˆ)14( ˆˆ

018104 and ˆ3ˆ2ˆ

zyx

zyx

kjizkjyixbar

nbkjin


7.9 Reciprocal vectors

The two sets of vectors and are called reciprocal sets ifcba

, , ''' c , ,

ba

'''

'''

''''''

'''

, , then vectorsunit orthogonal mutually are and , if

coplanarnot are and , ,0)( where

)(

)(

)(

0

1

ccbbaacba

cbacba

cba

bac

cba

acb

cba

cba

bcaccbabcaba

ccbbaa

Ex: Construct the reciprocal vector of kickjbia ˆˆ and ˆˆ ,ˆ2

kjkjic

jikib

kjikikja

kikjicba

ˆˆ2/)ˆˆ(ˆ2

ˆ2/ˆ2)ˆˆ(

2/)ˆˆˆ(2/)ˆˆ()ˆˆ(

2)]ˆˆ()ˆˆ[(ˆ2)(

'

'

'


Define the components of a vector with respect basis vectors that are not mutually orthogonal.

a

321ˆ and ˆ ,ˆ eee

3'32

'21

'1

'33

'22

1'133

'122

'111

'1

332211

'3

'2

'1

321

ˆ)ˆ(ˆ)ˆ(ˆ)ˆ(

ˆ ˆ

ˆˆˆˆˆ

ˆˆˆ

and , isvector basis

reciprocal its , and , basis lorthonorma-non Foe (2)

ˆ)ˆ(ˆ)ˆ(ˆ)ˆ(

ˆ and ˆ ,ˆvector baisi CartesianFor (1)

eeaeeaeeaa

eaaeaa

aeeaeeaeeaea

eaeaeaa

eee

eee

kkajjaiiaa

kji

�

7.1 Scalars and vectors Scalar: a quantity specified by its magnitude, for example: temperature,...

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Transcript of 7.1 Scalars and vectors Scalar: a quantity specified by its magnitude, for example: temperature,...