Day 23 in Mat228

Today:

• Announcements
  • Homework for 11.7 due today
  • Exam revisions graded: come in to discuss it and get 100% on the redo. Email me to schedule a meeting (Please use subject line REDO).
  • Short quiz on 13.1/13.2 today.
  • Next week (Thursday) we'll have our second exam. It will cover the rest of chapter 11 (11.2-11.7).

• Any questions on old stuff?

• Sections 13.1 and 13.2 basically define and treat vectors in two- and three-dimensional spaces.
  • Uniform Circular Motion is best understood using the concepts of vector geometry.

    So are many other concepts from physics, where quantities have a direction and a "strength" -- a rocket in flight, for example. It is moving in a certain direction, and it has a speed, whose value can be represented by the length of a vector.

    • vector: a quantity with length and direction:
      

    • In uniform circular motion, we could represent the position of the bob by and
    • The three vectors that are (most) important in the physics of circular motion are the vectors of
      • position,
      • velocity, and
      • acceleration,
      each of which is a function of time.
    • Animations, demonstrations:
      • Uniform Circular Motion
      • For all of the gory details, a nice layout of all the physics, and a cool (but scary) example, let's check out physclips -- the Australian dialect makes it more fun, too!

  • If a vector is in two-dimensions, then the length is just given using the Pythagorean theorem formula:
    
    

    in which case

    

    Obviously this can be generalized to vectors in four dimensions, five dimensions; even six dimensions. Maybe seven dimensions, too; or even n dimensions.

  • The vectors above are given with components:
    

    examples:

    • #5, p. 663
    • #9, p. 663
    • #11, p. 663
    
    

  • Figure 11 illustrates the so-called parallelogram rule:
    

    example:

    • #17, p. 664

    Here it is in three-space:

    

    with the obvious changes to formulas because you now have three components, instead of two:

    

  • Vectors with length 1 are special, called unit vectors, and are important because they can be dedicated to showing direction:

    It's clear that we can turn any vector into a unit vector, by simply scaling it:

    

    example:

    • #38, p. 664

  • We use special components and the parallelogram rule to write each vector in terms of a basis of vectors (equal in number to the dimension of the space): for example, in two-space, the usual basis is given by the two (perpendicular) vectors denoted i and j:
    

    and, in three-space,

    

  • Theorems:

    

    

    The Triangle inequality basically says that the diagonal of a parallelogram is shorter than or equal to the sum of the two sides:

    "The shortest distance between two points is a straight line."

    In the case of Figure 17 (for vectors at right angles to each other)

    this rule is familiar in the form of the old algebraic inequality
    

    and the only time you'll have equality is when either a or b is zero.

  • Let's look at some more examples:
    • #19, p. 664
    • #29, p. 664
    • #47, p. 664
    • More physics:
      • #58, p. 665
        • First and foremost: Newton's laws of motion (in particular ), and statics.
        • The parallelogram rule
        • We can think of the force vectors on the cables as being composed of vector components (as well as coordinate components).
      • #60, p. 665
      • Let's use a little calculus and the parameterization of the position vector in uniform circular motion given in one of our videos to derive the velocity and acceleration vectors, and check the video's claims.

• Additional material from section 13.2: three-dimensional vectors
  • The right-hand rule
    

  • Vocabulary:
    • coordinate planes (p. 666)
    • octants
    • translates/translation (p. 669)
    • parallel vectors
    • equivalent vectors
    • standard basis vectors

  • Distance formula in R³:
    

    examples:

    • #1, p. 673
    • #2, p. 673
    • #14, p. 673

  • Equations of Spheres and Cylinders

    What are a sphere and a right cylinder?

    • A sphere: all points equidistant from a given center.
    • A right cylinder: all points equidistant from a given line.
    

    examples: Draw the following:

    • The set of points satisfying

      

    • The set of points satisfying

      

  • Parametric Equations of Lines:

    

    

  • Additional examples:
    • #27, p. 674
    • #39, p. 674
    • #51, p. 674