Transcript Chap.2 Vectors - rossetto@univ

2. Vectors
 Geometric definition
D
1 - Modulus (length) > 0 : AB = AB
2 - Support (straight line): D,
or every straight line parallel to D
B
D’
A
D
C
3 - Direction (arrow)
Consequence: if CD = AB
if D’ // D
and if the orientation is the same
then:
CD = AB
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2. Vectors
 Algebric expression
y
V = vx ux  vy uy
vy
V
vx , vy : components

uy
0
V
x
ux
vx

v x  V cos 
vy  V sin 


vx

  0x, V ,mod.2
V
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vy

3
0    2 : if

then vx  0
2
2
if     2 then vy  0
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2. Vectors
 Definitions of operations on vectors
B
C
1 - Addition (Chasles relationship)
AB + BC = AC
The addition confers to the set of vectors
a structure of commutative group
( 0 is the neutral element
V the opposite element)
A
2 – Multiplication by a real number k
Distributivity/addition: k  , k(V1  V2)  kV1  kV2
These 2 operations confer to the set of vectors a structure
of commutative ring (k=1 is the neutral element)
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2. Vectors
 Dot product


V1  V2  V1 . V2 cos V1, V2  V1 OH
V1
H
1 – Geometric definition


   V1, V2 (commutativity)
V2
V2

N. B.: V  V  V
2
 0 square of the norm
2 - Orthonormality relationship
0
3 – Algebric expression
0 if m  n
um  un  mn  
1 if m=n
V1  V2  v1x v2x  v1y v2y  v1z v2z
VV  V
2
2
2
 v2

v

v
x
y
z (Pythagore Theorem)
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2. Vectors
 Properties of the dot product
1 – Commutativity: V1  V2  V2  V1
2 – Bilinearity: ,   , (V1  V2 )  U  V1  U  V2  U
,   , V  (U1  U2 )  V  U1  V  U2
Generalization : any scalar product A,B is a bilinear form defined on
 Properties of the norm
1 V
2
0
2  Schwartz inequality: A  B  A  B
Generalization: any norm is defined from these properties
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2. Vectors
 Dot product: other notations
a1 
 
A
1 - Matrix:
is a matrix with one column and 3 rows:  A   a2 
a 
 3
b1 
 
b2 
b 
 3
T
A  B   A  B  a1 a2 a3 
3
A B 
 ab
i i
i 1
 a1b1  a2b2  a3b3
 aubu
2 - Einstein convention: implicit sum on repeated indices
A=
3
 ai ui
i=1
 av uv , v=(1,2,3), u1 , u2 , u3 : coordinate system
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2. Vectors
 Using Einstein convention
Total differential
d   v  duv
 

 
dx 
dy 
dz 

y
z 
 x
Product of matrices
cij  ABij  aikbkj
n


   aikbkj 
 k 1

Trace of a matrix
Tr A  aii
 n

   aii 
 i 1 
ii  3
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2. Vectors
 Cross or vector product
V1  V2
1 – Geometric definition

V2


 V1  V2  V1 . V2 sin V1, V2


Direction: screw or right hand rules

V1

V1  V2  area V1  V2

N.B.: uu  uv  uw , u, v, w : permutation of x, y, z
ux  ux  uy  uy  uz  uz  0
2 – Properties: - anticommutativity: V1  V2   V2  V1
- bilinearity
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2. Vectors
 Cross product: other notations
3 – Algebric expression:

V1  V2  ux v1y v2z - v1z v2y

 uy  v1z v2x - v1x v2z 

 uz v1x v2y - v1y v2x

4 – Einstein convention: Levi-Civita symbol


A  B ijk uiajbk      uiajbk 
 i j k



0, unless i, j, k are distinct (21 cases of zero)

ijk +1 if (i,j,k) are (1,2,3) in cyclic order, or even permutation (3 cases)
-1 if (i,j,k) are (2,1,3) in cyclic order, or odd permutation (3 cases)

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2. Vectors
 On Levi-Civita symbol
1 – ijk  1 if i,j,k   1,2,3 ,  2,3,1 , 3,1,2 
ijk  1 if i,j,k   1,3,2 , 2,1,3 , 3,2,1
2 - Property: rotating indices doesn’t change sign:
ijk kij
3 - Component # i of the dot product


 A  B ijk ajbk     ijk ajbk 

i
 j k



4 - Relationship between Levi Civita and Kronecker symbols
ijkilm  jlkm  jmlk
Proof: examine the 81 cases and group symetric ones.
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2. Vectors
 Cross or vector product: computation
ux

V1  V2  det v1x

v2x


v1y v1z 

v2y v2z 
uy
uz
Sarrus rule:
a11

det  a21
a
 31
a12
a13
a11
a22
a23
a21
a32
a33
a31
a12
 a11
 
a22   a21
a32  a31
a12
a13
a11
a22
a23
a21
a32
a33
a31
a12


a22 
a32 
v1y v1z 
v1x v1y 
v1x v1z 
or V1  V2  ux det 
  uy det 

  uz det 
v2y v2z 
v2x v2y 
v2x v2z 
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2. Vectors
 Double cross product
a  (b  c)  b(a  c)  c(a  b) (bac – cab or abacab rule)
Pr oof. a  (b  c) ijk aj(b  c)k ijk aj klm bl cm
i
In order to apply the relationship between Levi-Civita and
Kronecker symbols, both Levi-Civita symbols have to begin
with the same indice k. Then we use the invariance by rotating
indices.
a  (b  c) ijk klm aj bl cm kijklm aj bl cm

i


 il jm  im jl aj bl cm  bi  amcm   ci  ab
l l
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2. Vectors
 Vector triple product


1 – Definition: A  B  C  ai ijk b jck
ax ay

2 – Expression: A  B  C  det bx by

cx cy

by
A  B  C  ax det 
cy






az 

bz 

cz 
bz 
bx
  ay det 
cz 
cx

bx
bz 
  az det 
cz 
cx

A  B  C  ax bycz - bzcy  ay bzcx - bxcz   az bxcy - bycx
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by 

cy 

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2. Vectors
 Properties of triple product





a bc  b ca  c  ab

Proof. Consider, for example, the first equation:



  
b   c  a  bx  cyaz - czay   by czax - cxaz   bz cxay - cyax 
a  b  c  ax by cz - bzcy  ay bzcx - bxcz   az bxcy - bycx
Other proof of the first equation using Levi-Civita symbols:




a  b  c  ai ijk bjck ijk ab
i jck jki ab
i jck  b j jki aick
We can permute a and c (but not indices) in Levi-Civita symbol


a  b  c  b j jki ck ai
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