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2. Vectors. Geometric definition. 1 - Modulus (length) > 0 : AB = 2 - Support (straight line): D, or every straight line parallel to D 3 - Direction (arrow). D. B. D’. A. Consequence: if CD = AB if D’ // D and if the orientation is the same

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2 vectors
2. Vectors

  • Geometric definition

1 - Modulus (length) > 0 : AB =

2 - Support (straight line): D,

or every straight line parallel to D

3 - Direction (arrow)

D

B

D’

A

Consequence: if CD = AB

if D’ // D

and if the orientation is the same

then:

D

C

B. Rossetto


2 vectors1
2. Vectors

  • Algebric expression

y

: components

q

x

0

q

q

B. Rossetto


2 vectors2
2. Vectors

  • Definitions of operations on vectors

1 - Addition (Chasles relationship)

B

C

The addition confers to the set of vectors

a structure of commutative group

( is the neutral element

the opposite element)

A

2 – Multiplication by a real number k

Distributivity/addition:

These 2 operations confer to the set of vectors a structure of commutative ring (k=1 is the neutral element)

B. Rossetto


2 vectors3
2. Vectors

  • Dot product

1 – Geometric definition

(commutativity)

H

q

2 - Orthonormality relationship

0

3 – Algebric expression

B. Rossetto


2 vectors4
2. Vectors

  • Properties of the dot product

1 – Commutativity:

2 – Bilinearity:

  • Properties of the norm

B. Rossetto


2 vectors5
2. Vectors

  • Dot product: other notations

1 - Matrix:

is a matrix with one column and 3 rows:

2 - Einstein convention: implicit sum on repeated indices

B. Rossetto


2 vectors6
2. Vectors

  • Using Einstein convention

Total differential

Product of matrices

Trace of a matrix

B. Rossetto


2 vectors7
2. Vectors

  • Cross or vector product

1 – Geometric definition

q

N.B.:

2 – Properties: - anticommutativity:

- bilinearity

B. Rossetto


2 vectors8
2. Vectors

  • Cross product: other notations

3 – Algebric expression:

4 – Einstein convention: Levi-Civita symbol

B. Rossetto


2 vectors9
2. Vectors

  • On Levi-Civita symbol

1 –

2 - Property: rotating indices doesn’t change sign:

3 - Component # i of the dot product

4 - Relationship between Levi Civita and Kronecker symbols

Proof: examine the 81 cases and group symetric ones.

B. Rossetto


2 vectors10
2. Vectors

  • Cross or vector product: computation

Sarrus rule:

B. Rossetto


2 vectors11
2. Vectors

  • Double cross product

(bac – cab or abacab rule)

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.

B. Rossetto


2 vectors12
2. Vectors

  • Vector triple product

1 – Definition:

2 – Expression:

B. Rossetto


2 vectors13
2. Vectors

  • Properties of triple product

Proof. Consider, for example, the first equation:

Other proof of the first equation using Levi-Civita symbols:

We can permute a and c (but not indices) in Levi-Civita symbol

B. Rossetto


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