Dave's Web Corner

MTH-261 Week 1 (August 25-31)

• Outline
• Assessments
• Handouts
• Video
• Technology

Outline

12.1 Three-Dimensional Coordinate Systems

• recommended exercises
  • 3, 7, 11, 13, 20, 29, 37, 43, 44
• learning objectives
  • describe three-dimensional space mathematically
  • locate points in space using coordinates
  • write the distance formula in three dimensions
  • write the equations for simple planes and spheres
• notes
  • the rectangular coordinate system
    • the coordinate planes
      • \( x = 0 \) (the \(yz\)-plane), \( y = 0 \) (the \(xz\)-plane), and \( z = 0 \) (the \(xy\)-plane)
    • planes (parallel to the coordinate planes)
      • \( x = a \), \( y = b \), and \( z = c \)
  • the right-hand rule
  • distance in 3-space
    • the natural extension of distance in the plane (2-space)
    • also can be extended into higher dimensions (n-space)
  • equations of spheres
    • \( x^{2}+y^{2}+z^{2}=r^{2} \)
    • \( (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2} \)

12.2 Vectors

• recommended exercises
  • 7, 8, 13, 17, 21, 35, 38, 40, 51
• learning objectives
  • describe a vector in the plane and in space, using correct notation
  • perform basic vector operations (scalar multiplication, addition, subtraction) geometrically and algebraically
  • express a vector in component form with respect to the standard unit vectors
  • express a vector as the product of its length and direction
• notes
  • the geometric description of vectors
    • initial point, terminal point
    • length (or magnitude), direction, and equality
    • scalar multiple, parallel vectors
    • addition and subtraction
    • the parallelogram law
  • vectors and components
    • components
      • \( \vec{v} = \langle v_{1},v_{2} \rangle = v_{1}\vec{\imath}+v_{2}\vec{\jmath} \)
      • \( \vec{v} = \langle v_{1},v_{2},v_{3} \rangle = v_{1}\vec{\imath}+v_{2}\vec{\jmath}+v_{3}\vec{k} \)
      • the vectors
        • \( \vec{\imath} = \langle 1,0 \rangle \) and \( \vec{\jmath} = \langle 0,1 \rangle \), or
        • \( \vec{\imath} = \langle 1,0,0 \rangle \), \( \vec{\jmath} = \langle 0,1,0 \rangle \), and \( \vec{k} = \langle 0,0,1 \rangle \)
        are the standard unit vectors in 2-space and 3-space respectively.
    • vector arithmetic
      • addition, subtraction, scaling, length and direction via components
    • unit vectors
      • any non-zero vector \( \vec{v} \), divided by its own length, is a unit vector, called the direction of \( \vec{v} \): \(\displaystyle{\frac{\vec{v}}{\|\vec{v}\|}}\)
    • every non-zero vector is a product of its own length and direction
      • \( \displaystyle{\vec{v}=\|\vec{v}\|\left(\frac{\vec{v}}{\|\vec{v}\|}\right)} \)
  • properties of vector operations
  • the resultant of two concurrent forces
  • vector arithmetic operations and properties can be extended to \( n \)-dimensional space

12.3 The Dot Product

• recommended exercises
  • 9, 11, 14, 17, 22, 29, 35, 38, 42, 54, 56, 61, 62, 63
• learning objectives
  • calculate the dot product of two vectors
  • determine whether two vectors are perpendicular
  • find the direction cosines of a given vector
  • calculate the vector projection of one vector onto another
  • calculate the work done by a given force
• notes
  • the dot product
    • the useful geometric description
    • the practical algebraic description
    • the scalar quantity work is defined naturally with a dot product
  • properties of the dot product
  • the Cauchy-Schwarz inequality and the triangle inequality
  • using the dot product to determine angle
    • generally
      • acute, obtuse, orthogonal (sometimes called perpendicular or normal)
    • specifically
      • using the inverse cosine
  • notice how the dot product interacts with the standard unit vectors
    • \( \vec{\imath}\cdot\vec{\imath} =\vec{\jmath}\cdot\vec{\jmath} =\vec{k}\cdot\vec{k} =1 \)
    • \( \vec{\imath}\cdot\vec{\jmath} =\vec{\jmath}\cdot\vec{k} =\vec{\imath}\cdot\vec{k} =0 \)
    • these can be verified both by direct calcluation and by the properties of the dot product
      • \( \vec{v}\cdot\vec{v}=\|\vec{v}\|^{2} \)
      • \( \vec{u}\cdot\vec{v}=0 \) when \( \vec{u} \) and \( \vec{u} \) are perpendicular
  • direction angles and direction cosines
  • orthogonal projection and orthogonal decomposition into parallel and normal components

12.4 The Cross Product

• recommended exercises
  • 15, 19, 28, 31, 33, 35, 45, 46, 47, 53
• learning objectives
  • calculate the cross product of two given vectors
  • use determinants to calculate a cross product
  • find a vector orthogonal to two given vectors
  • determine areas and volumes by using the cross product
  • calculate the torque of a given force and position vector
• notes
  • the practical algebraic description
    • motivation: creating a vector perpendicular to two independent non-zero vectors
    • matrices and determinants
    • the anticommutativity consequence
  • the logical geometric description
    • the right-hand rule
  • notice how the cross product interacts with the standard unit vectors
    • \( \vec{\imath}\times\vec{\imath} =\vec{\jmath}\times\vec{\jmath} =\vec{k}\times\vec{k} =\vec{0} \)
    • \( \vec{\imath}\times\vec{\jmath}=\vec{k} \), \( \vec{\jmath}\times\vec{k}=\vec{\imath} \), and \( \vec{k}\times\vec{\imath}=\vec{\jmath} \)
    • \( \vec{\jmath}\times\vec{\imath}=-\vec{k} \), \( \vec{k}\times\vec{\jmath}=-\vec{\imath} \), and \( \vec{\imath}\times\vec{k}=-\vec{\jmath} \)
    • these can be verified both by direct calcluation and by the properties of the cross product
      • \( \vec{v}\times\vec{v}=\vec{0} \)
      • the right-hand rule
      • \( \vec{u}\times\vec{v}=-\vec{v}\times\vec{u} \)
  • properties of the cross product
  • the cross product conveniently calculates
    • the area of a parallelogram, \( \|\vec{u}\times\vec{v})\| \), and related triangles
  • the cross product and the dot product combined conveniently calculate
    • the volume of a parallelepiped, \( |\vec{u}\cdot(\vec{v}\times\vec{w})| \), and related tetrahedra, \( |\vec{u}\cdot(\vec{v}\times\vec{w})|/6 \)
    • the combination \( \vec{u}\cdot(\vec{v}\times\vec{w}) \) is called the scalar triple product (or triple scalar product, or box product)
  • applications of the cross product
    • torque - the "work impersonator"

Assessments

• Deadlines
  • All homework is due at the beginning of class on the date indicated.
  • All exams are given in class on the date indicated.
  • Late homework is not accepted and there are no make-ups for missed exams. Contact me before the deadline if you have a conflict with a deadline and we will find a mutually agreeable solution.
  • Solutions are posted under these links after the deadline.
• Homework
  • due Thursday, August 29:
    • (none)
  • due Thursday, September 5:
    • 1, 2, 3, 4, 5, 6, 7, 8
  • due Thursday, September 12:
    • 9, 10, 11
• Exams
  • Thursday, October 3: Exam 1
    • Exam 1 (solutions)
    • Exam 1 covers sections 12.1-12.6, 13.1-13.4, and 14.1-14.3

Handouts

• Announcements
  • Using Mathematica.pdf
  • Schmidt Award
  • Math Graduate Award
  • Honors Program introduction
  • Tutoring Center services
• Graph Paper
  • Graph Paper is a folder containing a variery of graph paper
  • Rectangular Grids 1 Large Coarse
  • Rectangular Grids 1 Large Medium
  • Rectangular Grids 3 Small
  • Rectangular Grids 6 Small
  • Isometric Grids 1 Large
  • Isometric Grids 3 Small
  • Isometric Grids 6 Small
• Excerpts of Text
  • chap12sec01.pdf
  • chap12sec02.pdf
  • chap12sec03.pdf
  • chap12sec04.pdf
• Formula Sheets
  • Formulas 1
• Additional Exam Practice
  • Useful Vector and Curve Practice Problems.zip
• Lines, Planes, and Distance
  • Dot and Cross Product Properties and Uses
  • Dot Product Consequences (solution)
  • cross product consequences (solution)
  • Distance or Intersection (solution)
  • More Distance Problems
  • Line and Plane Equations
  • Parallelepiped Surgery
  • Parallelepiped Decomposed Into Six Tetrahedra

Video

• Recommended Problems
  • Chapter 2
    • 2-2-95, 2-2-119, 2-3-157, 2-3-181
  • all MTH-261 Recommended Problems
• Topics
  • The Cauchy-Schwarz Inequality
  • The Triangle Inequality
  • The Direction Cosines
  • all MTH-261 Topics

Technology

• Communication
  • Zoom
  • Joining a Zoom Webinar
• Mathematica
  • Basic Graphing.nb
  • Lines and Curves in Space.nb
  • Dot and Cross Products.nb
  • Packing Bowling Ball.nb
  • 261-chap12-sec01-exer44.nb