vector algebra
TRANSCRIPT
![Page 1: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/1.jpg)
Vector Algebra
![Page 2: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/2.jpg)
Course Content
I. Introduction to the CourseII. Biomechanical Concepts Related
to Human MovementIII. Anatomical Concepts Related to
Human MovementIV. Applications in Human
Movement
![Page 3: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/3.jpg)
Biomechanical Concepts
A. Basic Kinematic ConceptsB. Vector AlgebraC. Basic Kinetic Concepts
![Page 4: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/4.jpg)
Vector Algebra
1. Introductory Concepts2. Vector Composition3. Vector Resolution
![Page 5: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/5.jpg)
Vector Algebra
1. Introductory Concepts2. Vector Composition3. Vector Resolution
![Page 6: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/6.jpg)
Vector Algebra: Introductory Concepts
a. Definitionsb. Vector representationc. Muscle force vectors
![Page 7: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/7.jpg)
Definitions What is vector algebra? What is a scalar quantity? What is a vector quantity?
![Page 8: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/8.jpg)
Vector Representation
-y
+z
+x-x
+y
0°
90°
180°
270°
= -40°
-y
+z
+x-x
+y
![Page 9: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/9.jpg)
Vector 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 Vector
![Page 10: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/10.jpg)
Examples of Vector Representations
Luttgens & Hamilton. (2001). Fig 10.1. p. 266.
Luttgens & Hamilton. (2001). Fig 10.1. p. 266.
![Page 11: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/11.jpg)
Vector Representation
![Page 12: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/12.jpg)
Muscle Force Vectors
Point of application
Direction Magnitude Line of force
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 13: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/13.jpg)
Muscle Force Vectors
Biceps brachii
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 14: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/14.jpg)
Muscle Force Vectors
Brachialis
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 15: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/15.jpg)
Muscle Force Vectors
Deltoid
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 16: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/16.jpg)
Muscle Force Vectors
Pectoralis major
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 17: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/17.jpg)
Muscle Force Vectors
Pectoralis major
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 18: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/18.jpg)
Muscle Force Vectors
Pectoralis minor
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 19: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/19.jpg)
Vector Algebra
1. Introductory Concepts2. Vector Composition3. Vector Resolution
![Page 20: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/20.jpg)
Vector Composition Process of determining a
resultant vector from two or more vectors
New vector called the resultant (R)
![Page 21: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/21.jpg)
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.
![Page 22: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/22.jpg)
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.
![Page 23: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/23.jpg)
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.
![Page 24: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/24.jpg)
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.
![Page 25: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/25.jpg)
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.
![Page 26: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/26.jpg)
Vector Composition: Graphical Solution (Chaining)
Vector P = 50 N
What 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.
![Page 27: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/27.jpg)
Order of chaining does not matter.
D
R
Hamilton & 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?
A
C
B
0°
70°
![Page 28: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/28.jpg)
The same R can be achieved from an infinite combination of vectors.
Hamilton & Luttgens. (2001). Fig 10.2. p. 267.
![Page 29: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/29.jpg)
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?
![Page 30: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/30.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-6. p. 64.
![Page 31: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/31.jpg)
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?
![Page 32: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/32.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-13. p. 69.
![Page 33: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/33.jpg)
Vector Algebra
1. Introductory Concepts2. Vector Composition3. Vector Resolution
![Page 34: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/34.jpg)
Vector Resolution Taking a resultant vector and
breaking it down into 2 or more component vectors
![Page 35: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/35.jpg)
There 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 + 6.5
![Page 36: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/36.jpg)
So, how do we know which components to resolve for?
2D (3D conceptually)
Orthogonal
![Page 37: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/37.jpg)
So, 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.
![Page 38: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/38.jpg)
Vector Resolution:Graphical Solution Draw a
rectangle which includes R as the diagonal of the rectangle.
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-33. p. 79.
![Page 39: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/39.jpg)
Hamilton & Luttgens. (2001). Fig 10.1. p. 266.
Why might you want to do this?
Vh or Vx
Vv or Vy
If Vr was 200 m/s, what is the magnitude of Vv and Vh?
![Page 40: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/40.jpg)
Hamilton & Luttgens. (2001). Fig 10.1. p. 266.
Vh or Vx
Vv or Vy
![Page 41: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/41.jpg)
Resolving Muscle Force Vectors
Direction of resolution is in direction of interest.
In this case, movement of shoulder girdle is vertical (elevation & depression) and horizontal (protraction & retraction).
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 42: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/42.jpg)
Resolving Muscle Force Vectors
1. Draw line of pull.2. Draw vertical
component.3. Draw horizontal
component.4. Complete rectangle to
assure proper magnitudes of components.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 43: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/43.jpg)
1. Draw line of pull.2. Draw vertical
component.3. Draw horizontal
component.4. Complete rectangle to
assure proper magnitudes of components.
What are the linear effects produced by this muscle?
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 44: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/44.jpg)
1. Draw line of pull.2. Draw vertical
component.3. Draw horizontal
component.4. Complete rectangle to
assure proper magnitudes of components.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 45: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/45.jpg)
1. Draw line of pull.2. Draw vertical
component.3. Draw horizontal
component.4. Complete rectangle to
assure proper magnitudes of components.
If the resultant force is 100 N, how much force is acting to elevate the scapula? To retract the scapula?
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 46: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/46.jpg)
Mechanical Axis of a Bone The longitudinal
axis of the bone
![Page 47: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/47.jpg)
Resolving Muscle Force Vectors
1. Draw a line to represent the mechanical axis of the bone.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 48: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/48.jpg)
Fnormal
2. Draw in the normal component first.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 49: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/49.jpg)
Fnormal
Ftangential
3. Draw in the tangential component second.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 50: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/50.jpg)
Fnormal
Ftangential
4. Complete the rectangle to make sure that you have the lengths of your component vectors correct.
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 51: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/51.jpg)
Fnormal
Ftangential
How would you express the direction of the resultant muscle force? The components?
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
0°
![Page 52: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/52.jpg)
Fnormal
Ftangential
What are the linear effects produced by this muscle?
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 53: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/53.jpg)
Fnormal
Ftangential
If the resultant muscle force is 500 N, what is the magnitude of the components?
Source: Mediclip. (1995). Baltimore: Williams & Wilkins.
![Page 54: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/54.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.
1. Draw a line to represent the mechanical axis of the bone.
![Page 55: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/55.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.
2. Draw in the normal component first.
Fnormal
![Page 56: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/56.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.
3. Draw in the tangential component second.
Ftangential
Fnormal
![Page 57: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/57.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.
4. Complete the rectangle to make sure that you have the lengths of your vectors correct.
Ftangential
Fnormal
![Page 58: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/58.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-30. p. 77.
Ftangential
Fnormal
How would you express the direction of the resultant muscle force? The components?
0
![Page 59: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/59.jpg)
Fnormal
Ftangential
Fnormal
Ftangential
Component magnitudes vary, depending on magnitude & direction of R.
![Page 60: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/60.jpg)
Vector Resolution: Other
Fw,parallel
Fw,perpendicular
![Page 61: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/61.jpg)
Fv
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-28. p. 75.
![Page 62: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/62.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-29. p. 76.
Differences in normal component?
![Page 63: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/63.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-29. p. 76.
Differences in tangential component?
Differences in muscle insertion angle?
![Page 64: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/64.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-31. p. 77.
![Page 65: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/65.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-32. p. 78.
![Page 66: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/66.jpg)
From: LeVeau, B.F. (1992). William & Lissner’s biomechanics of human motion (3rd ed). Philadelphia: W.B. Saunders. Fig. 4-36. p. 82.
![Page 67: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/67.jpg)
Value of Vector Analysis Helps us understand forces and
their effects!
![Page 68: Vector Algebra](https://reader033.vdocument.in/reader033/viewer/2022061108/544df240b1af9f17508b494b/html5/thumbnails/68.jpg)
For the next lecture day: Lecture Topic #2
Subtopic C – Basic Kinetic Concepts