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Derivation of the Vector Dot Product and the Vector Cross Product

Derivation of the Vector Dot Product and the Vector Cross Product

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Derivation of the Vector Dot Product and the Vector Cross Product

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  1. Derivation of the Vector Dot Product and the Vector Cross Product

  2. Derivation of the Vector Dot Product u·v =∑i ui vi = ∑i ui ei ∑i vj ej

  3. (u1 e1+u2 e2+u3 e3) (v1 e1+v2 e2+v3 e3) Kronecker Delta ei·ej = δij = 1 when i = j 0 when i ≠ j

  4. = u1 e1v1 e1+ u1 e1v2 e2+u1 e1v3e3+u2 e2v1 e1+u2 e2v2 e2+u2 e2v3 e3+u3 e3 v1 e1+u3 e3 v2 e2+u3 e3 v3 e3

  5. = u1v1e1e1+ u2v2e2e2+u3v3e3e3 = u1v1+ u2v2+u3v3

  6. Vector Cross Product Einstein Notation u × υ = εijk e i ujυk = Σijkεijkeiujυk = Σi Σj Σk εijkeiujυk

  7. Levi-Civati Symbol 0 unless i, j, k are distinct +1 if i, j, k is an even permutation of (1, 2, 3) -1 if i. j, k is an odd permutation of (1, 2, 3) ε =

  8. Derivation of the Cross Product =(ε121u2v1+ ε122 u2v2+ ε123u2v3 + ε131u3v1+ ε132u3v2 + ε133 u3v3) e1+ (ε211u1v1+ ε212u1v2+ ε213u1v3 + ε231u3v1 + ε232 u3v2+ ε233 u3v3)e2+ (ε311 u1v1+ ε312 u1v2 + ε313 u1v3+ ε321 u2v1 + ε322 u2v2+ ε323 u2v3) e3

  9. Levi-Civati Symbol • 3 1 3 1 2 2 • even 123, 231, 312 odd 321, 213, 132

  10. Derivation of the Cross Product =(ε123u2v3+ ε132u3v2) e1 + (ε213 u1v3 + ε231 u3v1) e2+ (ε312u1v2 + ε321 u2v1) e3 = (u2v3 – u3v2)e1 + (u1v3 – u3v1)e2 + (u1v2 – u2v1)e3