Vector Math for 3D Computer Graphics. Interactive Tutorial

Local Disk......Page 0
Vector Math Tutorial for 3D Computer Graphics......Page 1
CHAPTER 1 --- Vectors, Points, and Column Matrices......Page 3
Virtual Tourist......Page 4
Vector......Page 6
Column Matrix......Page 8
Variables as Elements......Page 10
Convenient Printing of Column Matrices......Page 11
Column Matrix Equality......Page 12
Matrices as Data Types......Page 14
Names for Matrices......Page 16
Column Matrices representing Vectors......Page 18
Displacement......Page 20
Reading Displacements from Graph Paper......Page 22
Checking an Answer......Page 24
Vectors Don't have a Location......Page 25
Subtracting Points 
No, I'm not talking about your midterm......Page 27
Practice with Displacements......Page 28
Real Numbers......Page 29
file:///C|/InetPub/wwwroot/VectorLessons/vch01/vch01_18.html......Page 30
CHAPTER 2 -- Column and Row Matrix Addition......Page 31
Transposition......Page 32
Column Matrix Addition......Page 33
Three Separate Problems......Page 34
Commutative......Page 35
Zero Column Matrix......Page 36
Matrix Addition is Associative......Page 37
Some things which Cannot be Done......Page 38
Matrix Subtraction......Page 39
Not Commutative......Page 40
Moving "-" Inside......Page 41
More Practice......Page 43
Yet More Practice......Page 44
Oh No! Algebra......Page 45
End of the Chapter......Page 47
CHAPTER 3 -- Vector Addition......Page 49
Vectors in 2D Space......Page 50
Position Independent......Page 51
Head-to-Tail Rule......Page 52
Vector Addition in 3D......Page 53
Associative......Page 54
Commutative......Page 55
Representing Points with Column Matrices......Page 56
Representing Points with Different Frames......Page 58
Representing Vectors with Column Matrices......Page 59
Vector Addition represented by
Column Matrix Addition......Page 60
Different Order, but Same Answer......Page 61
Practice......Page 62
Same Problem......Page 63
Parallelogram for Vector Addition......Page 64
More Practice......Page 65
Two Legged Trip......Page 66
Summing Displacements =/= Summing Lengths......Page 67
Vectors in the Same Direction......Page 68
Diagram for Adding the Zero Vector......Page 69
Adding Points and Vectors......Page 70
Translation......Page 71
End of Chapter......Page 72
CHAPTER 4 -- Vector Length......Page 73
Vectors Aligned with X-Y Axes......Page 74
Pythagorean Formula......Page 75
Formula for Vector Length......Page 77
Symbol for Vector Length......Page 78
Demonstration the Triangle Inequality......Page 79
Another Example......Page 81
Length of 3D Vectors......Page 82
Geometrical Vectors......Page 83
More Practice......Page 84
Length is always Positive......Page 86
Length of the Negative of a Vector......Page 87
Opposite Direction......Page 88
End of this Lengthy Chapter......Page 89
Vector Lessons Chapter 5 -- Vector Direction......Page 90
Vectors Aligned with X-Y Axes......Page 91
Angle from the X Axis......Page 92
Watch those "-" Signs......Page 94
More Meddlesome Minuses......Page 95
Calculating both Attributes......Page 96
Length and Direction converted to (x, y)......Page 97
Radians......Page 98
Practice with Radians......Page 100
Lost Bugs......Page 101
Buggy Directions......Page 103
End of this Lesson......Page 104
CHAPTER 6 -- Scaling and Unit Vectors......Page 105
Scaling......Page 106
A few more fun Facts......Page 108
Answer is always a Vector......Page 109
Expressions with Mixed Types......Page 110
More Algebra!......Page 111
Equating Corresponding Elements......Page 113
Scaling with values less than One......Page 115
Unit Vector......Page 116
Creating Unit Vectors......Page 117
Direction of a Unit Vector......Page 118
Check on the Answer......Page 119
Opposite Direction......Page 120
End of this Chapter......Page 121
CHAPTER 7 -- Dot Product......Page 122
The Dot Product......Page 123
Commutative......Page 125
Cosine of 90°......Page 126
Dot Product of Orthogonal Vectors......Page 127
Pure Orthogonality Detector......Page 128
Properties of Dot Product......Page 129
Dot Product of Column Matrices......Page 130
Same Properties......Page 131
Commutative......Page 132
Mixed Types == Trouble......Page 133
Dot Product in Three Dimensions......Page 134
Dot Product with Zero Vector......Page 136
Any Real is Possible......Page 137
Alway Positive......Page 138
Distributive with repect to Vector Addition......Page 139
Demonstration of Distributive Property......Page 140
Practice with Distribution......Page 141
End of the Chapter......Page 142
CHAPTER 8 -- Length, Orthogonality, and the Column Matrix Dot Product......Page 143
Dot Product Formula for Length......Page 144
Practice......Page 145
Unit Vectors......Page 146
Keep the length Explicit......Page 147
Unit Vectors as Directions......Page 148
Right Angles......Page 150
Same Vectors, Rotated......Page 151
Dot Product of Orthogonal Vectors......Page 152
Independent of Length......Page 153
Infinitely Many Normals......Page 155
Zero Vector as Orthogonal......Page 156
Slopes of Perpendicular Lines......Page 158
End of the Chapter......Page 159
CHAPTER 9 --- The Angle Between Two Vectors......Page 160
Example Dot Products......Page 161
Length Affects Dot Product......Page 162
Normalize Vectors......Page 163
Filtering out the effect of Length......Page 164
Range of the Dot Product of Two Unit Vectors......Page 166
Formula for the Angle between Two Vectors......Page 168
Practice......Page 169
More Practice......Page 171
Which Side?......Page 172
Angles Greater than 90......Page 173
Same Angle; Same Dot Product......Page 174
More Practice......Page 175
Non-unit Vectors......Page 176
file:///C|/InetPub/wwwroot/VectorLessons/vch09/vch09_15.html......Page 177
End of the Chapter......Page 178
CHAPTER 10 --- Angle between 3D Vectors......Page 179
file:///C|/InetPub/wwwroot/VectorLessons/vch10/vch10_2.html......Page 180
Angles between Coordinate Axes......Page 181
Angle between two Vectors......Page 182
Formula for the angle between two Vectors......Page 183
A More Difficult Problem......Page 184
Applying the Formula......Page 185
Angle between Two Directions......Page 186
More Practice......Page 188
The Answer Worked Out......Page 189
Geometry......Page 190
Answer Worked Out......Page 191
End of the Chapter......Page 192
CHAPTER 11 --- Projections......Page 193
file:///C|/InetPub/wwwroot/VectorLessons/vch11/vch11_2.html......Page 194
Two Vectors Define a Plane......Page 195
Projection of one Vector onto Another......Page 196
Solving for kv......Page 197
Projection Procedure......Page 199
Finding u......Page 200
Example......Page 201
Another Example......Page 203
The Point of All This......Page 205
Check the Answer......Page 206
Groan! Another Example  will they never end...........Page 207
Eyeball Estimate......Page 208
Example in Three Dimensions......Page 209
Collapsing one Dimension......Page 210
Another Example......Page 212
Checking the Answer......Page 213
End of the Chapter......Page 215
CHAPTER 12 --- Vector Cross Product......Page 216
file:///C|/InetPub/wwwroot/VectorLessons/vch12/vch12_2.html......Page 217
file:///C|/InetPub/wwwroot/VectorLessons/vch12/vch12_3.html......Page 218
Carpentry of Vectors......Page 219
Visualizing the Cross Product......Page 220
Reversing the Order of Operands......Page 221
Colinear Operands......Page 222
More Math Fun......Page 223
Not Associative......Page 224
More Properties......Page 225
Computing Cross Product from Column Matrices......Page 227
Memory Aid......Page 228
Evaluating the Determinant-like Thing......Page 230
Another Evaluation......Page 232
Two Rows Equal......Page 233
End of Chapter......Page 234
file:///C|/InetPub/wwwroot/VectorLessons/vmch13/vmch13_1.html......Page 235
Definition of Matrix......Page 236
Matrix Dimensions......Page 237
Square Matrix......Page 238
CHAPTER 14 --- Matrix times Column Matrix......Page 239
Matrix times Column Matrix......Page 240
More Practice......Page 242
Column Matrix on the Right, only......Page 243
CHAPTER 15 --- Product of Rectangular Matrices......Page 244
Dimensions of the Result......Page 245
Practice with Dimensions......Page 246
Forming the Matrix-Matrix Product......Page 247
CHAPTER 16 --- Identity Matrix and Matrix Inverse......Page 249
Square Matrices......Page 250
Pre-multiplication by I......Page 251
Identity Matrix......Page 252
Vectors and Matrices Index......Page 253
Matrix Addition......Page 260
Zero Matrix......Page 261
Rules for Matrix Addition......Page 262
Practice with Matrix Addition......Page 263
Flip the Column......Page 264
More Practice......Page 266
Dot Product as Matrix Multiplication......Page 268
Determinant of a Singular Matrix......Page 270
Rank of a Matrix......Page 271
Matrix Equality......Page 272
I is Unique......Page 273
Transpose of I......Page 275
file:///C|/InetPub/wwwroot/VectorLessons/vmch16/vmch16_7.html......Page 277
Matrix Inverse......Page 278
Non-singular......Page 279
Properties of the Matrix Inverse......Page 280
Associative......Page 281
Distributive Property......Page 283
Another Way to Visualize......Page 286
Summary of Rules (so far)......Page 288
Not Commutative......Page 289
More Practice......Page 290
Scalar Factor......Page 291
Row Matrix times Matrix
 as Several Dot Products......Page 292
Row Matrix always on the Left......Page 294
Unique A-1......Page 295
Repeated Dot Products......Page 297
More Flipping Practice......Page 298
More Practice......Page 300
End of the Chapter......Page 302
Steps in Multiplication......Page 303
End of the Chapter......Page 304
Matrix Subtraction......Page 305
Transpose......Page 306
file:///C|/InetPub/wwwroot/VectorLessons/vmch14/vmch14_12.html......Page 308
A Rule for Transpose......Page 309
Product Yields I......Page 310
Names for Matrices......Page 311
Negative of a Matrix......Page 312
Multipliation of a Matrix by a Scalar......Page 313
Multiplication Practice......Page 314
Rule Summary......Page 316
Solve for p......Page 318
Real Number Elements......Page 320
End of the Chapter......Page 322