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Chapter 4: Vector Geometry

Chapter 4: Vector Geometry

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Chapter 4: Vector Geometry
4.1 Vectors and Lines
Vectors in \mathbb { R } ^ { 3 }
Length and Direction
Theorem 4.1.1
Proof. Let v have point P ( x , y , z ) .
Example 4.1.1
Theorem 4.1.2
Geometric Vectors
Definition 4.1 Geometric Vectors
The Parallelogram Law
The Parallelogram Law
Example 4.1.2
Theorem 4.1.3
Theorem 4.1.4
Example 4.1.3
Scalar Multiple Law
Proof.
Example 4.1.4
Example 4.1.5
Example 4.1.6
Definition 4.2 Parallel Vectors in R3
Theorem 4.1.5
Example 4.1.7
Lines in Space
Definition 4.3 Direction Vector of a Line
Vector Equation of a Line
Parametric Equations of a Line
Example 4.1.8
Example 4.1.9
Example 4.1.10
Example 4.1.11
Pythagoras’ Theorem
Theorem 4.1.6: Pythagoras’ Theorem
Exercises for 4.1
Exercise 4.1.3
Exercise 4.1.31
4.2 Projections and Planes
The Dot Product and Angles
Definition 4.4 Dot Product in R3
Example 4.2.1
Theorem 4.2.1
Example 4.2.2
Law of Cosines
Theorem 4.2.2
Example 4.2.3
Definition 4.5 Orthogonal Vectors in \mathbb{R}^3
Theorem 4.2.3
Example 4.2.4
Example 4.2.5
Projections
Example 4.2.6
Definition 4.6 Projection in R3
Theorem 4.2.4
Example 4.2.7
Example 4.2.8
Planes
Definition 4.7 Normal Vector in a Plane
Scalar Equation of a Plane
Example 4.2.9
Example 4.2.10
Vector Equation of a Plane
Example 4.2.11
The Cross Product
Definition 4.8 Cross Product
Determinant Form of the Cross Product
Example 4.2.12
Theorem 4.2.5
Example 4.2.13
Example 4.2.14
Exercises for 4.2
Exercise 4.2.17
Exercise 4.2.26
Exercise 4.2.32
Exercise 4.2.34
Exercise 4.2.39
Exercise 4.2.44
4.3 More on the Cross Product
Theorem 4.3.1
Theorem 4.3.2
Theorem 4.3.3: Lagrange Identity12
Theorem 4.3.4
Example 4.3.1
Theorem 4.3.5
Example 4.3.2
Right-hand Rule
Exercises for 4.3
Exercise 4.3.16
4.4 Linear Operators on \mathbb { R } ^ { 3 }
Definition 4.9 Linear Operator on Rn
Theorem 4.4.1
Reflections and Projections
Theorem 4.4.2
Theorem 4.4.3
Rotations
Example 4.4.1
Transformations of Areas and Volumes
Theorem 4.4.4
Theorem 4.4.5
Proof.
Exercises for 4.4
Exercise 4.4.4
4.5 An Application to Computer Graphics
Example 4.5.1
Exercises for 4.5
Supplementary Exercises for Chapter 4