Day 47 in MAT229

Last time

Next time

Today:

Announcements:
- You have a homework due today (parts 4 and 5 of the worksheet).
- Quiz opportunity today
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 ${x(t)=R\cos(t)}$ and ${y(t)=R\sin(t)}$
    We then combine these into one quantity, the vector ${\langle{x(t),y(t)}\rangle}$ .
  - The three vectors that are (most) important in the physics of circular motion are the vectors of
    - position ${\langle{x(t),y(t)}\rangle}$ ,
    - velocity ${\langle{x'(t),y'(t)}\rangle}$ , and
    - acceleration ${\langle{x''(t),y''(t)}\rangle}$ ,
    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
  ${\overline{v}=\langle{a,b}\rangle}$
  
  Hence in 3-d,
  we have
  
  ${||\overline{PQ}||=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2+(c_2-c_1)^2}}$
  or for ${\overline{v}=\langle{a,b,c}\rangle}$
  
  ${||\overline{v}||=\sqrt{a^2+b^2+c^2}}$
  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
  
  ${(x-h)^2+(y-k)^2+(z-l)^2=r^2}$
  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
  
  ${\sqrt{a^2+b^2}\le{|a|+|b|}}$
  and the only time you'll have equality is when either a or b is zero.
Examples:
- #29, p. 823
- #31
- #42, p. 824

Website maintained by Andy Long. Comments appreciated.