4: Vector Geometry
- Page ID
- 58854
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Vectors are presented intrinsically in terms of length and direction, and are related to matrices via coordinates. Then vector operations are defined using matrices and shown to be the same as the corresponding intrinsic definitions. Next, dot products and projections are introduced to solve problems about lines and planes. This leads to the cross product. Then matrix transformations are introduced in \(\mathbb{R}^3\), matrices of projections and reflections are derived, and areas and volumes are computed using determinants. The chapter closes with an application to computer graphics.
- 4.1: Vectors and Lines
- This chapter covers the geometry of 3-dimensional space, focusing on points as vectors in a Cartesian coordinate system. It explains fundamental geometric properties of vectors, vector addition using the parallelogram law, and distance calculations in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). The text also introduces weighted averages, midpoints, and the concept of parallel vectors.
- 4.2: Projections and Planes
- This page covers key concepts in geometry related to vectors, including perpendicularity, the dot product, projections, and the cross product. It explains how to determine angles and orthogonality through the dot product, illustrates vector projections in three dimensions, and introduces equations for planes. The properties of the cross product, including its orthogonal nature and its application in defining planes and calculating distances, are emphasized.
- 4.3: More on the Cross Product
- This page covers the cross product of vectors in \(\mathbb{R}^3\), emphasizing its properties, geometric interpretation, and computational techniques. It explains that the cross product yields a vector orthogonal to two given vectors and discusses its linearity, behavior with zero or equal vectors, and the Lagrange Identity.
- 4.4: Linear Operators on R³
- This page covers linear transformations in \(\mathbb{R}^3\), focusing on linear operators such as rotations, reflections, and projections. It defines linearity, provides matrix representations for various transformations, and discusses properties of isometries. The relationships between reflections and projections are explored, along with their geometric implications.
- 4.5: An Application to Computer Graphics
- This page covers the fundamentals of computer graphics, focusing on image creation through points and instructions for curves, along with matrix transformations to maintain linearity. It provides examples of displaying the letter 'A' and applying transformations like shear and scaling using homogeneous coordinates.
- 4.E: Supplementary Exercises for Chapter 4
- This page provides supplementary exercises for Chapter 4 focused on vector concepts and geometry. It includes proofs of unique coefficients in vector combinations, discussions on triangle medians and altitudes' concurrency, and practical problems involving swimmer and airplane velocities in currents. Key topics emphasize geometric relationships and vector calculations, including determining distances from points to planes and analyzing resultant speeds and directions.