Vector AlgebraCourse ContentBiomechanical ConceptsSlide 4Slide 5Vector Algebra: Introductory ConceptsDefinitionsVector RepresentationSlide 9Examples of Vector RepresentationsSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Vector CompositionVector Composition: Graphical Solution (Chaining)Slide 22Slide 23Slide 24Slide 25Slide 26Order of chaining does not matter.The same R can be achieved from an infinite combination of vectors.Magnitude of R is dependent on direction of components, not just magnitude.Slide 30Slide 31Slide 32Slide 33Vector ResolutionThere is an infinite # of combinations of component vectors for any given R.So, how do we know which components to resolve for?Slide 37Vector Resolution: Graphical SolutionSlide 39Slide 40Resolving Muscle Force VectorsSlide 42Slide 43Slide 44Slide 45Mechanical Axis of a BoneSlide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Vector Resolution: OtherSlide 61Slide 62Slide 63Slide 64Slide 65Slide 66Value of Vector AnalysisFor the next lecture day:Vector AlgebraCourse ContentI. Introduction to the CourseII. Biomechanical Concepts Related to Human MovementIII. Anatomical Concepts Related to Human MovementIV. Applications in Human MovementBiomechanical ConceptsA. Basic Kinematic ConceptsB. Vector AlgebraC. Basic Kinetic ConceptsVector Algebra1. Introductory Concepts2. Vector Composition3. Vector ResolutionVector Algebra1. Introductory Concepts2. Vector Composition3. Vector ResolutionVector Algebra: Introductory Conceptsa. Definitionsb. Vector representationc. Muscle force vectorsDefinitionsWhat is vector algebra?What is a scalar quantity?What is a vector quantity?Vector Representation-y+z+x-x+y0°90°180°270° = -40°-y+z+x-x+yVector RepresentationA vector quantity is represented by an arrow.Arrow head represents direction.Tail represents point of forceapplication.Line of force (or pull).Length represents magnitude.Force VectorExamples of Vector RepresentationsLuttgens & Hamilton. (2001). Fig 10.1. p. 266.Luttgens & Hamilton. (2001). Fig 10.1. p. 266.Vector RepresentationMuscle Force VectorsPoint of applicationDirectionMagnitudeLine of forceSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsBiceps brachiiSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsBrachialisSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsDeltoidSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsPectoralis majorSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsPectoralis majorSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Muscle Force VectorsPectoralis minorSource: Mediclip. (1995). Baltimore: Williams & Wilkins.Vector Algebra1. Introductory Concepts2. Vector Composition3. Vector ResolutionVector CompositionProcess of determining a resultant vector from two or more vectorsNew vector called the resultant (R)Vector Composition: Graphical Solution (Chaining)From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.1. Select a vector to start with and draw it, maintaining direction and magnitude.Vector Composition: Graphical Solution (Chaining)From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.2. Chain the tail of the next vector to the head of the first, maintaining direction and magnitude from original vector.Vector Composition: Graphical Solution (Chaining)From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.3. Continue to chain vectors in this manner until they are all chained.Vector Composition: Graphical Solution (Chaining)From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.4. Draw in the resultant vector by connecting the tail of the first vector in the chain to the head of the last vector in the chain.Vector Composition: Graphical Solution (Chaining)5. The head of the resultant vector will be the end that is connected to the head of the last vector.From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.Vector Composition: Graphical Solution (Chaining)Vector P = 50 NWhat is the magnitude of the resultant vector?From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-3. p. 63.Order of chaining does not matter.DRHamilton & Luttgens. (2001). Fig 10.2. p. 267.If A=50 N of force, what would you estimate the magnitude of R to be?How would you state the direction of R?ACB0°70°The same R can be achieved from an infinite combination of vectors.Hamilton & Luttgens. (2001). Fig 10.2. p. 267.Magnitude of R is dependent on direction of components, not just magnitude.If F=300 N of force, what would you estimate the magnitude of R to be?How would you state the direction of R?From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-6. p. 64.From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-12. p. 69.If Q=50 N of force, what would you estimate the magnitude of R to be?How would you state the direction of R?From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-13. p. 69.Vector Algebra1. Introductory Concepts2. Vector Composition3. Vector ResolutionVector ResolutionTaking a resultant vector and breaking it down into 2 or more component vectorsThere is an infinite # of combinations of component vectors for any given R.8 = 4 + 48 = 3 + 1 + 2 + 28 = 10 + (-2)8 = 1.5 + 6.5So, how do we know which components to resolve for?2D (3D conceptually)OrthogonalSo, how do we know which components to resolve for?2D (3D conceptually)OrthogonalHorizontal & VerticalExceptionsMusclesOther From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-33. p. 79.Vector Resolution:Graphical SolutionDraw a rectangle which includes R as the diagonal of the