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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

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2. Vectors

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  1. 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. 2. Vectors • Algebric expression y : components q x 0 q q B. Rossetto

  3. 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

  4. 2. Vectors • Dot product 1 – Geometric definition (commutativity) H q 2 - Orthonormality relationship 0 3 – Algebric expression B. Rossetto

  5. 2. Vectors • Properties of the dot product 1 – Commutativity: 2 – Bilinearity: • Properties of the norm B. Rossetto

  6. 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

  7. 2. Vectors • Using Einstein convention Total differential Product of matrices Trace of a matrix B. Rossetto

  8. 2. Vectors • Cross or vector product 1 – Geometric definition q N.B.: 2 – Properties: - anticommutativity: - bilinearity B. Rossetto

  9. 2. Vectors • Cross product: other notations 3 – Algebric expression: 4 – Einstein convention: Levi-Civita symbol B. Rossetto

  10. 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

  11. 2. Vectors • Cross or vector product: computation Sarrus rule: B. Rossetto

  12. 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

  13. 2. Vectors • Vector triple product 1 – Definition: 2 – Expression: B. Rossetto

  14. 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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