Derivation of the Vector Dot Product and the

Derivation of the Vector Dot Product and the Vector Cross Product

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

(u 1 e 1 + u 2 e 2 + u 3 e 3) (v 1 e 1 + v 2 e 2 + 3 v e 3) Kronecker Delta ei·ej = δij = 1 when i = j 0 when i ≠ j

= u 1 e 1 v 1 e 1 + u 1 e 1 v 2 e 2 + u 1 e 1 v 3 e 3 2 1 2 2 2 3 + u e 2 v e 1 + u e 2 v e 2 + u e 2 v e 3 + u 3 e 3 v 1 e 1 + u 3 e 3 v 2 e 2 + u 3 e 3 v 3 e 3

= u 1 v 1 e 1 e 1+ u 2 v 2 e 2 e 2+u 3 v 3 e 3 e 3 = u 1 v 1+ u 2 v 2+u 3 v 3

Vector Cross Product Einstein Notation j k u × υ = εijk e i u υ = Σijkεijkeiujυk = Σi Σj Σk εijkeiujυk

Levi-Civati Symbol 0 unless i, j, k are distinct ε = +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)

Derivation of the Cross Product = (ε 121 u 2 v 1 + ε 122 u 2 v 2 + ε 123 u 2 v 3 + ε 131 u 3 v 1 + ε 132 u 3 v 2 + ε 133 u 3 v 3 ) e 1+ (ε 211 u 1 v 1 + ε 212 u 1 v 2 + ε 213 u 1 v 3 + ε 231 u 3 v 1 + ε 232 u 3 v 2 + ε 233 u 3 v 3 )e 2 + (ε 311 u 1 v 1 + ε 312 u 1 v 2 + ε 313 u 1 v 3 + ε 321 u 2 v 1 + ε 322 u 2 v 2 + ε 323 u 2 v 3 ) e 3

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

Derivation of the Cross Product = (ε 123 u 2 v 3+ ε 132 u 3 v 2) e 1 + (ε 213 u 1 v 3 + ε 231 u 3 v 1) e 2+ (ε 312 u 1 v 2 + ε 321 u 2 v 1) e 3 = (u 2 v 3 – u 3 v 2)e 1 + (u 1 v 3 – u 3 v 1)e 2 + (u 1 v 2 – u 2 v 1)e 3