4: Coordinate Geometry in Three Dimensions
- Page ID
- 6811
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Introduction
- This page discusses three-dimensional geometry, focusing on rectangular, cylindrical, and spherical coordinates. It highlights right-handed and left-handed coordinate systems, and explains how equations for x, y, and z represent surfaces, requiring multiple equations for lines or curves. Six types of surfaces are examined: plane, ellipsoid, paraboloid, hyperboloid, cylinder, and cone.
- 4.2: Planes and Straight Lines
- This page covers the geometry of planes and triangles in three-dimensional space, explaining the plane equation \(Ax + By + Cz + D = 0\) and methods for determining intersections and distances. It describes deriving plane equations from points, calculating areas of triangles and volumes of tetrahedrons using determinants, and conditions for the intersection of three planes.
- 4.3: The Ellipsoid
- This page explores the mathematical equations of spheroids and ellipsoids, highlighting oblate and prolate forms related to the Earth's shape and focusing on their properties and implications. It also details how to represent an ellipse from an ellipsoid intersection, including angle transformations and area calculations, supplemented by a practical example and a Fortran code to compute cross-sectional areas in various directions.
- 4.4: The Paraboloid
- This page covers different types of parabolic surfaces in the \(xz\)-plane and their 3D forms, starting with the standard parabola's properties, such as vertex and focus. It explains how rotating a parabola creates circular paraboloids, often used in telescopes. Additionally, it details elliptic and hyperbolic paraboloids, emphasizing their unique cross-sectional shapes, defining equations, and geometric characteristics in relation to the Cartesian plane.
- 4.5: The Hyperboloid
- This page explores hyperbolas and hyperboloids, detailing how rotating a hyperbola around axes affects its geometry. It explains that the equation \(\frac{x^2}{a^2} - \frac{z^2}{c^2} = 1\) yields a circular hyperboloid of one sheet when rotated around the \(z\)-axis and one of two sheets around the \(x\)-axis. Furthermore, it introduces other hyperbola equations, encouraging readers to visualize different cross-sections.
- 4.6: The Cylinder
- This page explains the equation for a cylinder with an elliptical cross-section, specifically \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), aligned with the \(z\)-axis. It also notes that variations in axes and orientations can be achieved through translation and rotation of coordinate axes, as previously discussed in the textbook.
- 4.7: The Cone
- This page covers the geometric representation and equations of cones, focusing on the equation \(x^2 + y^2 = a^2 z^2\), which describes a cone with its vertex at the origin and vertical axis. It explains the semi-vertical angle \(α\), elliptical cross-sections, and deriving equations for cones with axes in arbitrary directions. Additionally, it highlights practical applications in astronomy, particularly the mathematical relationships related to meteoroid shock fronts.
- 4.8: The General Second Degree Equation in Three Dimensions
- This page explains second-degree equations in three dimensions, covering various surfaces like planes and ellipsoids. It highlights that nine independent constants define these surfaces and discusses special cases regarding their position relative to the origin. An example equation illustrates how to simplify the equation using coordinate transformations.
- 4.9: Matrices
- This page covers key matrix operations, including vector and matrix multiplication, determinants, matrix inversion, diagonalization of symmetric matrices, and orthogonality testing. It emphasizes the importance of mastering these operations manually for better comprehension of their arithmetic and geometric interpretations, despite the availability of computer tools. The section also provides tests for matrix orthogonality and encourages visualization to aid understanding.