Optimized entropy-constrained vector quantization of lossy vector map compression
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
description
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 - PowerPoint PPT Presentation
Transcript of 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 ui vi
= ∑i ui ei ∑i vj ej
(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
= u1v1e1e1+ u2v2e2e2+u3v3e3e3
= u1v1+ u2v2+u3v3
Vector Cross Product Einstein Notation
u × υ = εijk e i uj υk
= Σijkεijkeiujυk
= Σi Σj Σk εijkeiujυk
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)
Derivation of the Cross Product
Levi-Civati Symbol
3 1 3 1 2 2
even 123, 231, 312 odd 321, 213, 132
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
