Derivation of the Vector Dot Product and the Vector Cross Product

Slides:

Advertisements

Similar presentations

General information about M7 repairing tools

Advertisements

Central Visual & Performing Arts High School

Central Visual & Performing Arts High School

Urban Morphological Zones v An improved dataset for urban analysis Roger Milego i Agràs ETC-TE Workshop on Land Accounts and Urban Morphology Bellaterra,

Macroeconomic Themes:31 Contribution of Monetarism in Macroeconomic Policy Supply of money is the determinant of the national income In the long run, the.

Macroeconomic Analysis 2003

L3: Texture Components and Euler Angles: part 2

Regular operations Sipser 1.1 (pages 44 – 47). CS 311 Fall Building languages If L is a language, then its complement is L’ = {w | w ∉ L} Let A.

Common Variable Types in Elasticity

God Alone.com Advocating the worship of God Alone.

Dual Credit NHS Requirements

This trick relies on three nice ideas from discrete mathematics: pigeonhole principle 5 pigeons (cards) in 4 holes (suits) means at least two pigeons are.

EXIT NEXT Click one of the buttons below or press the enter key BACKTOPICS Probability The MEnTe Program Math Enrichment through Technology Title V East.

Thermodynamics and Statistical Mechanics First Law of Thermodynamics.

Find a minimum-state FA recognizing the same language as the given FA b b a a a a a a a b b b b b List all unordered.

Non-Deterministic Finite Automata

Applications of Inclusion-Exclusion: Selected Exercises.

De-Duplication A not so simple problem Covers Appendix Part 5.

Hard Problems Some problems are hard to solve. No polynomial time algorithm is known Most combinatorial optimization problems are hard Popular NP-hard.

Common Variable Types in Elasticity

Chapter_12.indd 13/17/11 5:16 PM. Chapter_12.indd 23/17/11 5:16 PM.

Chapter_15.indd 13/17/11 5:25 PM. Chapter_15.indd 23/17/11 5:25 PM.

Chapter_11.indd 13/17/11 5:11 PM. Chapter_11.indd 23/17/11 5:11 PM.

Chapter_3.indd 13/17/11 4:30 PM. Chapter_3.indd 23/17/11 4:30 PM.

Global Trade and International Classifications.

Geometry of R2 and R3 Dot and Cross Products.

P m -3 m {100} P-surface. P m -3 m {211} P m -3 m {100}+{111} Neovius surface.

Coordinates. Basis  A basis is a set of elements that generate a group or field. Groups have a minimum set that generates the group.Groups have a minimum.

Tensors. Jacobian Matrix  A general transformation can be expressed as a matrix. Partial derivatives between two systemsPartial derivatives between two.

Summer School 2007B. Rossetto1 2. Vectors  Geometric definition 1 - Modulus (length) > 0 : AB = 2 - Support (straight line): D, or every straight line.

Vectors. Vector Space  A vector space is an abstract mathematical object.  It has a set of vectors. Commutative with additionCommutative with addition.

AppxA_01fig_PChem.jpg Complex Numbers i. AppxA_02fig_PChem.jpg Complex Conjugate.

1, A.J.Mcconnell, applications of tensor analysis, dover publications,Inc , NEW York 2, Garl E.pearson, handbook of applied mathematics, Van Nostrand.

Chapter 1 Vector analysis

Powerful tool For Effective study and to Understand Flow Devices…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Vector Notation.

Boundar Layer Meteorology Lecture 2 Review chapter 1 of Garratt Terminology and Notation Review Some non-dimensional numbers Reynolds Averaging.

(a) (b) (c) (d). What is (1,2,3)  (3,4,2)? (a) (1, 2, 3, 4) (b) (1,2)  (3,4) (c) (1,3,4,2) (d) (3,1)  (4,2)

Vectors. We will start with a basic review of vectors.

Tutorial: Calculation of Image Rotation for a Scanning Optical System Sergio Guevara OPTI 521 – Distance Learning College of Optical Science University.

1 CEE 451G ENVIRONMENTAL FLUID MECHANICS LECTURE 1: SCALARS, VECTORS AND TENSORS A scalar has magnitude but no direction. An example is pressure p. The.

Lecture 3: Tensor Analysis a – scalar A i – vector, i=1,2,3 σ ij – tensor, i=1,2,3; j=1,2,3 Rules for Tensor Manipulation: 1.A subscript occurring twice.

1. Vector Analysis 1.1 Vector Algebra Vector operations A scalar has a magnitude (mass, time, temperature, charge). A vector has a magnitude (its.

Factorial Notation For any positive integer n, n! means: n (n – 1) (n – 2)... (3) (2) (1) 0! will be defined as equal to one. Examples: 4! = =

Basic biomechanics and bioacoustics

Powerful tool For Effective study and to Understand Flow Devices…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Tensor Notation.

Lecture 2 Mathematical preliminaries and tensor analysis

Scalars & Vectors in Terms of Transformations: Sect. 1.8 Skip Sects. 1,5, 1.6, 1.7 for now. Possibly come back to later (as needed!) Consider the orthogonal.

Dr. Shildneck. Dot Product Again, there are two types of Vector Multiplication. The inner product,called the Dot Product, and the outer product called.

§ Linear Spaces Christopher Crawford PHY

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

Inequality Notation.

tensor calculus 02 - tensor calculus - tensor algebra.

Warm up 1.) (3, 2, -4), (-1, 0, -7) Find the vector in standard position and find the magnitude of the vector.

Christopher Crawford PHY

Application of Vector product

Tuesday, February 8.

Copy and paste function can then be applied

Copy and paste function can then be applied

Continuum Mechanics (MTH487)

Basic biomechanics and bioacoustics

Principles of Computing – UFCFA3-30-1

1.2 Finding Limits Graphically and Numerically

 Unique leftmost derivation  Unique rightmost derivation num  Unique leftmost derivation  Unique rightmost derivation.

Coma / Virgo Border A1 12h30m +18° +16° +14° +12° +10° +8° 12h40m

Dots and Cross Products of Vectors in Space

Copy and paste function can then be applied

Copy and paste function can then be applied

Copy and paste function can then be applied

Presentation transcript:

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

= 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

= 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) ε =

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

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