1 / 10

Derivation of the Vector Dot Product and the Vector Cross Product

Derivation of the Vector Dot Product and the Vector Cross Product. Derivation of the Vector Dot Product. u·v =∑ i u i v i = ∑ i u i e i ∑ i v j e j. (u 1 e 1 + u 2 e 2 + u 3 e 3 ) (v 1 e 1 + v 2 e 2 + v 3 e 3 ). Kronecker Delta e i ·e j = δ ij = 1 when i = j

lirit
Télécharger la présentation

Derivation of the Vector Dot Product and the Vector Cross Product

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related