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Intro
Text Organization
- The text is divided into 10 chapters.
- In Search of sections within each chapter encourage students to investigate and explore challenging topics individually, or in small groups.
- Making Connections pages relate mathematics to other fields and disciplines, and encourage a greater understanding of the nature and purpose of mathematics.
- The problem supplement at the end of the text provides an opportunity for students to synthesize the skills and ideas acquired throughout the course.
- The answer key provides answers for all exercises, reviews, and inventories, as well as for the problem supplement.
- The glossary provides definitions of relevant mathematical terms.
- The index lists topics and main concepts for easy reference.
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Chapter One
In the first seven chapters of this book, you will be studying the algebra of mathematical objects called vectors.
Vectors are among the most recent inventions in mathematics that you will encounter in high school. Indeed, vector analysis was only fully developed at the turn of the 20th century. Unfortunately, very few properties and theorems in vector analysis are named after their originators. To compensate for this, a short history of the development of vectors is presented here.
1.1 What is a Vector?
1.2 Three-dimensional space
In Search of Trigonometry as an Aid to Visualization of 3-Space
1.3 Vectors as Ordered Pairs of Tripples
1.4 Vector Addition
Making Connections: Between Pigeons and Problem Solving
1.5 Properties of Vector Addition
1.6 Vector Subtraction
1.7 Multiplication of a Vector by a Scalar
1.8 Applications of Vector Subtraction and Multiplcation by a Scalar
1.9 Unit Vectors — Standard Basis of a Vector Space
In Search of Vectors as Classes of an Equivalence Relation
Summary
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Chapter Two
In this chapter you will learn more about the linear combinations of vectors and about the dependence of vectors on each other.
2.1 Linear Dependence of Two Vectors
In Search of a Solution for a System with Three Variables: Elimination
2.2 Linear Dependence of Three Vectors
2.3 Linearly Independent Vectors and Basis Vectors
In Search of Vectors in Spaces with Dimension Higher than Three
Making Connections: The Prisoners' Dilemma — a Game
2.4 Points of Division of a Line Segment
In Search of a Solution for a System with Three Variables: Matrices
2.5 Collinear Points and Coplanar Points
2.6 Geometric Proofs Using Linear Independence of Vectors
Summary
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Chapter Three
Once you have learned to multiply vectors, you will appreciate that vector analysis is a very powerful tool that brings 3-space geometric problems to a level hardly more difficult than problems in 2-space. You will be using products of vectors extensively in the rest of your work on vectors in this book.
3.1 Projections and Components
3.2 The Dot Product
3.3 Properties of the Dot Product
3.4 Applications: The Dot Product and Trigonometry
3.5 The Cross Product
Making Connections: Bridges
3.6 Properties of the Cross Product
Summary
- Chapter Four
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Chapter Five
5.1 The Vector Equation of a Line
5.2 Parametric Equations of a Line
Making Conections: Sailboards and Sailboats
5.3 Cartesian Equations of Lines
5.4 Direction Numbers and Direction Cosines of a Line
5.5 The Intersection of Lines in 2-Space
5.6 The Intersection of Lines in 3-Space
5.7 Geometric Proofs Using Vector Equations
In Search of a Proof of Desargues' Theorem
Summary
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Chapter Six
6.1 Vector and Parametric Equations of a Plane in 3-Space
6.2 The Cartesian Equation of a Plane in 3-Space
Making Conections: Lines and Planes
6.3 The Intersection of Lines and Planes
6.4 The Intersection of Two Planes
In Search of Vector Equations of Circles and Spheres
6.5 The Intersection of Three Planes
6.6 The Distance from a Point to a Plane or to a Line
Summary
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Chapter Seven
In this chapter, you will be introduced to a new mathematical object called a matrix (plural matrices) that can be used as an operator to effect various transformations.
Matrices provide a very compact way of expressing transformations. Recall that many vector equations can be applied to 2-space, 3-space, and even to spaces of higher dimensions. Matrix equations also have this same universality.
This chapter will provide you with an extensive study of transformations of 2-space, but the principles that you learn will be readily applicable to transformations of 3-space.
7.1 Matrices
7.2 Matrices and Linear Transformations
7.3 The Effect of a Linear Transformation
7.4 Rotations and Reflections
7.5 Inverse Transformations
7.6 Composition of Transformations and Multiplication of Matrices
Making Connections: Similarity and Folding
7.7 Properties of Matrix Multiplication and Matrix Equations
In Search of Invariant Lines and Characteristic Vectors
Summary
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Chapter Eight
In this chapter you will study the transformation of conics under translations and rotations.
8.1 Conics in Standard Position
8.2 Translations of Conics
8.3 Translating Conics into Standard Position
8.4 Matrices, their Transposes and the Central Conics
8.5 Rotations of Central Conics
Making Connections: Earth Satelites
8.6 Rotations that Eliminate xy Terms
In Search of a Method of Eliminating the xy Term using Characteristic Values
Summary
- Chapter Nine
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Chapter Ten
10.1 What is a Complex Number?
10.2 Operations in ℂ
10.3 Geometric Representation of a Complex Number
!0.4 Equations in ℂ
10.5 Complex Numbers and Trigonometry
10.6 De Moivre's Theorem
10.7 Quest for Roots in ℂ
10.8 Graphing and Complex Numbers
In Search of Graphical Representation of Non-real Solutions of Equations
10.9 Exponential Form of a Complex Number
In Search of Other Results using the Exponential Form
Making Connections: The Visual Display of Data
Summary
- Supplement