Download presentation

Presentation is loading. Please wait.

Published byDavis Duddleston Modified over 3 years ago

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 = when i = j 0 when i ≠ j

4
= 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

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

6
**Vector Cross Product Einstein Notation**

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

7
**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 =

8
**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

9
Levi-Civati Symbol even 123, 231, odd 321, 213, 132

10
**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

Similar presentations

OK

An alphabet is any finite set of symbols. Examples: ASCII, Unicode, {0,1} ( binary alphabet ), {a,b,c}. Alphabets.

An alphabet is any finite set of symbols. Examples: ASCII, Unicode, {0,1} ( binary alphabet ), {a,b,c}. Alphabets.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google