Vector Algebra

Vector Algebra

Course Content

I. Introduction to the CourseII. Biomechanical Concepts Related

to Human MovementIII. Anatomical Concepts Related to

Human MovementIV. Applications in Human

Movement

Biomechanical Concepts

A. Basic Kinematic ConceptsB. Vector AlgebraC. Basic Kinetic Concepts

Vector Algebra

1. Introductory Concepts2. Vector Composition3. Vector Resolution

Vector Algebra


Vector Algebra: Introductory Concepts

a. Definitionsb. Vector representationc. Muscle force vectors

Definitions What is vector algebra? What is a scalar quantity? What is a vector quantity?

Vector Representation

-y

+z

+x-x

+y

0°

90°

180°

270°

= -40°

-y

+z

+x-x

+y

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

Examples of Vector Representations

Luttgens & Hamilton. (2001). Fig 10.1. p. 266.

Luttgens & Hamilton. (2001). Fig 10.1. p. 266.

Vector Representation

Muscle Force Vectors

Point of application

Direction Magnitude Line of force

Source: Mediclip. (1995). Baltimore: Williams & Wilkins.


Biceps brachii



Brachialis



Deltoid



Pectoralis major



Pectoralis major



Pectoralis minor


Vector Algebra


Vector 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.



2. Chain the tail of the next vector to the head of the first, maintaining direction and magnitude from original vector.



3. Continue to chain vectors in this manner until they are all chained.



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.


5. The head of the resultant vector will be the end that is connected to the head of the last vector.



Vector P = 50 N

What is the magnitude of the resultant vector?


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°

The same R can be achieved from an infinite combination of vectors.


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?



If Q=50 N of force, what would you estimate the magnitude of R to be?

How would you state the direction of R?


Vector Algebra


Vector Resolution Taking a resultant vector and

breaking it down into 2 or more component vectors

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

So, how do we know which components to resolve for?

2D (3D conceptually)

Orthogonal

So, how do we know which components to resolve for?

2D (3D conceptually)

Orthogonal Horizontal &

Vertical Exceptions

Muscles Other


Vector Resolution:Graphical Solution Draw a

rectangle which includes R as the diagonal of the rectangle.



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?


Vh or Vx

Vv or Vy

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).



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?











If the resultant force is 100 N, how much force is acting to elevate the scapula? To retract the scapula?


Mechanical Axis of a Bone The longitudinal

axis of the bone


1. Draw a line to represent the mechanical axis of the bone.


Fnormal

2. Draw in the normal component first.


Fnormal

Ftangential

3. Draw in the tangential component second.


Fnormal

Ftangential

4. Complete the rectangle to make sure that you have the lengths of your component vectors correct.


Fnormal

Ftangential

How would you express the direction of the resultant muscle force? The components?


0°

Fnormal

Ftangential

What are the linear effects produced by this muscle?


Fnormal

Ftangential

If the resultant muscle force is 500 N, what is the magnitude of the components?



1. Draw a line to represent the mechanical axis of the bone.


2. Draw in the normal component first.

Fnormal


3. Draw in the tangential component second.

Ftangential

Fnormal


4. Complete the rectangle to make sure that you have the lengths of your vectors correct.

Ftangential

Fnormal


Ftangential

Fnormal

How would you express the direction of the resultant muscle force? The components?

0

Fnormal

Ftangential

Fnormal

Ftangential

Component magnitudes vary, depending on magnitude & direction of R.

Vector Resolution: Other

Fw,parallel

Fw,perpendicular

Fv



Differences in normal component?


Differences in tangential component?

Differences in muscle insertion angle?




Value of Vector Analysis Helps us understand forces and

their effects!

For the next lecture day: Lecture Topic #2

Subtopic C – Basic Kinetic Concepts

Documents

Vector Algebra