This tutorial was copied to this server with permission from Dr. Bradley P. Kjell on 01/28/08.  The original was published here.  Also see here and here for more material by Dr. Kjell and information about Dr. Kjell.

Index by chapter only

Topical index follows

• CHAPTER 0 -- Points and Lines
  • Picture of a Point
  • Physical Points
  • Points in Space
  • Lines
  • Very Thin Wire
  • Triangles
  • Representation
  • Coordinates
  • Little Boxes
  • More Points
  • The Point and its Representation
  • Another Frame
  • Sunlight
  • Vector
  • Representing a Vector
  • Vectors and Frames
  • End of the Chapter

• CHAPTER 1 -- Vectors, Points, and Column Matrices
  • Virtual Tourist
  • Complete Picture
  • Another Virtual Snapshot
  • Vector
  • Column Matrix
  • Variables as Elements
  • Convenient Printing of Column Matrices
  • Matrix Equality
  • Matrices as Data Types
  • Names for Matrices
  • Column Matrices representing Vectors
  • Displacement
  • Reading Displacements from Graph Paper
  • Checking an Answer
  • Vectors Don't have a Location
  • Subtracting Points
  • Practice with Displacements
  • Real Numbers
  • End of Chapter

• CHAPTER 2 - Column and Row Matrix Addition
  • Transposition
  • Column Matrix Addition
  • Three Separate Problem
  • Commutative
  • Zero Column Matrix
  • Matrix Addition is Associative
  • Some things which Cannot be Done
  • Matrix Subtraction
  • Not Commutative
  • Moving "-" Inside
  • More Practice
  • Yet More Practice
  • Algebra
  • End of the Chapter

• CHAPTER 3 - Vector Addition
  • Vectors in 2D Space
  • Position Independent
  • Head-to-Tail Rule
  • Vector Addition in 3D
  • Associative
  • Commutative
  • Representing Points with Column Matrices
  • Representing Points with Different Frames
  • Representing Vectors with Column Matrices
  • Vector Addition represented by
    Column Matrix Addition
  • Different Order, but Same Answer
  • Practice
  • Same Problem
  • Parallelogram for Vector Addition
  • More Practice
  • Two Legged Trip
  • Summing Displacements =/= Summing Lengths
  • Vectors in the Same Direction
  • Diagram for Adding the Zero Vector
  • Adding Points and Vectors
  • Translation
  • End of Chapter

• CHAPTER 4 - Vector Length
  • Vectors Aligned with X-Y Axes
  • Pythagorean Formula
  • Formula for Vector Length
  • Symbol for Vector Length
  • Demonstration the Triangle Inequality
  • Another Example
  • Length of 3D Vectors
  • Geometrical Vectors
  • More Practice
  • Length is always Positive or Zero
  • Length of the Negative of a Vector
  • Opposite Direction
  • End of this Lengthy Chapter

• CHAPTER 5- Vector Direction
  • Vectors Aligned with X-Y Axes
  • Angle from the X Axis
  • Watch those "-" Signs
  • More Meddlesome Minuses
  • Calculating both Attributes
  • Length and Direction converted to (x, y)
  • Radians
  • Practice with Radians
  • Lost Bugs
  • Buggy Directions
  • End of this Lesson

• CHAPTER 6 - Scaling and Unit Vectors
  • Scaling
  • A few more fun Facts
  • Answer is always a Vector
  • Expressions with Mixed Types
  • More Algebra!
  • Equating Corresponding Elements
  • Scaling with values less than One
  • Unit Vector
  • Creating Unit Vectors
  • Direction of a Unit Vector
  • Check on the Answer
  • Opposite Direction
  • End of this Chapter

• CHAPTER 7 - Dot Product
  • The Dot Product
  • Commutative
  • Cosine of 90°
  • Dot Product of Orthogonal Vectors
  • Orthogonality Detector and Length Calculator
  • Properties of Dot Product
  • Review of Dot Product Properties
  • Dot Product of Column Matrices
  • Same Properties
  • Commutative
  • Mixed Types == Trouble
  • Dot Product in Three Dimensions
  • Dot Product with Zero Vector
  • Any Real is Possible
  • Always Positive
  • Distributive with respect to Vector Addition
  • Demonstration of Distributive Property
  • Practice with Distribution
  • End of the Chapter

• CHAPTER 8 - Length, Orthogonality, and the Column Matrix Dot Product
  • Dot Product Formula for Length
  • Practice
  • Unit Vectors
  • Keep the length Explicit
  • Unit Vectors as Directions
  • Right Angles
  • Same Vectors, Rotated
  • Dot Product of Orthogonal Vectors
  • Independent of Length
  • Infinitely Many Normals
  • Zero Vector as Orthogonal
  • Slopes of Perpendicular Lines
  • End of the Chapter

• CHAPTER 9 - The Angle Between Two Vectors
  • Example Dot Products
  • Length Affects Dot Product
  • Normalize Vectors
  • Filtering out the effect of Length
  • Range of the Dot Product of Two Unit Vectors
  • Formula for the Angle between Two Vectors
  • Practice
  • More Practice
  • Which Side?
  • Angles Greater than 90
  • Same Angle; Same Dot Product
  • More Practice
  • Non-unit Vectors
  • Angle between Non-unit Vectors
  • End of the Chapter

• CHAPTER 10 - Angle between 3D Vectors
  • Angles in Three Dimensions
  • Angles between Coordinate Axes
  • Angle between two Vectors
  • Formula for the angle between two Vectors
  • A More Difficult Problem
  • Applying the Formula
  • Angle between Two Directions
  • More Practice
  • The Answer Worked Out
  • Geometry
  • Answer Worked Out
  • End of the Chapter

• Chapter 11 - Projections
  • Many 2D Projections
  • Two Vectors Define a Plane
  • Projection of one Vector onto Another
  • Solving for kv
  • Projection Procedure
  • Finding u
  • Example
  • Another Example
  • The Point of All This
  • Check the Answer
  • Groan! Another Example
  • Eyeball Estimate
  • Example in Three Dimensions
  • Collapsing one Dimension
  • Another Example
  • Checking the Answer
  • End of the Chapter

• Chapter 12 - Vector Cross Product
  • zz
  • zz

• Chapter 13 - Matrices and Simple Matrix Operations
  • zz
  • zz

• Chapter 14 - Matrix times Column Matrix
  • zz
  • zz

• Chapter 15 - Product of Rectangular Matrices
  • zz
  • zz

• Chapter 16 - Identity Matrix and Matrix Inverse
  • zz
  • zz

Alphabetical Index

File:  KjellVectorTutorialIndex.htm

Revised:  03/05/08