Day 48 in MAT229

Today:

• Announcements:

  • Your homework (worksheets) returned. Graded
    • 4b
    • 4e

  • Quiz opportunity tomorrow will be over 12.3

• Section 12.1: Three-dimensional coordinate systems and Section 12.2: Vectors
  • 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

      We then combine these into one quantity, the vector .

    • 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:
    

    For a vector given by

    

    
    

    Hence in 3-d,

    we have

    

    or for

    

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

    Examples:

    • 3-D Point drawing instructions
    • #1, p. 814
    • #2
    • #3
    • #4
    • #5
    • #6

    The equation for a sphere centered at the point (h,k,l) of radius r given in our text is

    

    This is the set of points P at a distance of r from the point (h,k,l).

    Examples:

    • #12, p. 814
    • #17
    • #21
    • #23-33 odd

  • The vectors above are given with components:
    
    
    

  • Figure 11 illustrates the so-called parallelogram rule:
    

    Examples:

    • #5, p. 822
    • #15

    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:

    

    Examples:

    • #23, p. 823
    • #24

  • 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.

  Examples:

  • #29, p. 823
  • #31
  • #42, p. 824