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**Derivation of the Vector Dot Product and the Vector Cross Product**

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**Derivation of the Vector Dot Product**

u·v =∑i ui vi = ∑i ui ei ∑i vj ej

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**(u1 e1 + u2 e2 + u3 e3) (v1 e1 + v2 e2 + v3 e3)**

Kronecker Delta ei·ej = δij = when i = j 0 when i ≠ j

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= u1 e1 v1 e1 + u1 e1 v2 e2 + u1 e1 v3e3 + u2 e2 v1 e1 + u2 e2 v2 e2 + u2 e2 v3 e3 + u3 e3 v1 e1 + u3 e3 v2 e2 + u3 e3 v3 e3

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

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**Vector Cross Product Einstein Notation**

u × υ = εijk e i uj υk = Σijkεijkeiujυk = Σi Σj Σk εijkeiujυk

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**Levi-Civati Symbol ε = 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 =

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**Derivation of the Cross Product**

= (ε121 u2v1 + ε122 u2v2 + ε123 u2v3 + ε131 u3v1 + ε132 u3v2 + ε133 u3v3 ) e1+ (ε211 u1v1 + ε212 u1v2 + ε213 u1v3 + ε231 u3v1 + ε232 u3v2 + ε233 u3v3 )e2 + (ε311 u1v1 + ε312 u1v2 + ε313 u1v3 + ε321 u2v1 + ε322 u2v2 + ε323 u2v3 ) e3

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Levi-Civati Symbol even 123, 231, odd 321, 213, 132

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**Derivation of the Cross Product**

= (ε123 u2v3+ ε132u3v2) e1 + (ε213 u1v3 + ε231 u3v1) e2+ (ε312 u1v2 + ε321 u2v1) e3 = (u2v3 – u3v2)e1 + (u1v3 – u3v1)e2 + (u1v2 – u2v1)e3

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