Skeletal Blog for Math 200, Section 102, Fall 2015

September 9: Classes start.  

Begin 12.1-12.3: Coordinates, Vectors, and the Dot Product in 
  Dimensions 2 and 3.

12.1: Coordinates, distance between two points, and what certain equations
look like, such as  x^2 + y^2 = 4 (a cylinder), the equation
(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 (a sphere; consider the 2-d equivalent
which discards the (z-l)^2 term and is a circle in the plane).
Tricky Exercises in 12.1: Exercises 15-18 (completing the square),
Midpoints (Ex 19) and Equation of a Sphere from one diameter (Ex 20).

KEY POINT(S) for 12.1: When things look difficult to understand in 3-dimensions,
look for 2-dimensional or even 1-dimensional analogue.  E.g.

x^2 + y^2 + z^2 = 4  is a sphere (of radius 2) in 3-dimensions; but
what about
x^2 - 2x + y^2 - 6y + z^2 = 100 ?
in 1-dimension the analogue would be something like
x^2 - 2x = 30,  for this we know to "complete the square":
(x-1)^2 = x^2 - 2x + 1, and hence (x-1)^2 = (x^2 - 2x ) + 1 = 30 + 1 =31
same idea works in 2-dimensions and 3-dimensions:  so given
x^2 - 2x + y^2 - 6y + z^2 = 100   , we complete the squares in x and y:
(x-1)^2 + (y-3)^2 + z^2 = x^2 - 2x + 1 + y^2 - 6y + 9 + z^2 =
  = (x^2 - 2x + y^2 - 6y + z^2) + 1 + 9 =  100 + 1 + 9 = 110.
So this becomes
(x-1)^2 + (y-3)^2 + z^2 = 110 , which is a sphere with centre (1,3,0)
  and radius square_root(110).

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September 11: Begin 12.2 and 12.3:

12.2 Vectors and notation: 
vectors = magnitude + direction 
vectors are "equivalent" or "equal" if they have the same magnitude and
  direction 
Displacement view: <2,4> means move +2 in x-direction, +4 in y-direction,
  so "from (10,0) move to (12,4)", "from (-8,100) move to (-6,104)", ...
"displacement vectors from point A to B," boldface v = AB with arrow over it
  has "initial point A and terminal point B"
Note: In class we often write vectors with arrows over them.
Vector addition by placing displacement vectors from point to point
  (gives the Parallelogram Law; draw a picture).
Algebraic treatment of vectors: use coordinates and set the initial point
  to the origin (as a default).  
  So <3,5> is the vector from O=<0,0> to A=<3,5> in the plane.
Magnitude or length is the distance between its initial and terminal point
  (which is independent of its representation).
Vectors can be "scaled" by any real number.
  So |<3,5>| = sqrt(3^2+5^2).  Similarly for 3-dimensional vectors.
Vectors can be defined in any dimension and have various properties
  (see bottom p. 819) such as  a+b = b+a (Parallelogram), 7(a+b)= 7 a + 7 b,
  etc.
Often use the vectors i,j,k to describe 3-dim sapce (p.820).  So
  <2,3,6> = 2 i + 3 j + 6 k  .
Unit vectors: extemely useful (for non-obvious reasons to come):
  u =  a / |a|  is a unit vector pointing in the direction of a.
Physics application: Find the forces acting on a hanging weight.

KEY POINT(S) 12.2: The applications and motivations of vectors can be quite
subtle: the "displacement" interpretation is very down to earth; the
"forces" interpretation is more subtle, but reasonably concrete: e.g.,
if a barbell is being acted upon by the forces of (1) gravity, (2) a
weight lifter, (3) a breeze in the gym, one adds the forces and applies
the formula:  
  F (the sum of the forces as vectors) = mass . acceleration
to see how the barbell will be accelerated as a result of the 3 forces.

Also: taking a vector and finding its "direction" by computing the
  associated unit vector is extremely useful in practice (e.g., to
  organize a data structure of 100 points in the plane or in 3-dimensional
  space).  This is also very useful in 12.5.

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12.3 Dot Product:
  < a_1,a_2> dot < b_1,b_2 > = a_1 b_1 + a_2 b_2 ; similarly for 3-dims.
Dot product has many simple linearity properties (see box 2, page 825).
Fundamental:  a dot b = |a| |b| cos(theta)   (see box 3, page 825).
Definition: two vectors are orthogonal (perpendicular) if their dot product
  is zero.
OMIT: Direction Angles: cute idea, but not particularly important; 
  this section is packed enough as is.
Projections are a crucial idea in "linear regression" that is used in most
  aspects of science, social sciences, humanities, music, etc. where 
  quantative data is analyzed.  Here we get an introduction.
proj_a(b) is the vector projection of  b  onto the line spanned by  a.  It 
  is given by:  proj_a(b) = ( ( a dot b) / |a|^2  ) a  
OMIT: "scalar projection of b onto a" or "component of b along a".
Work is informally force applied over a distance or displacement; more
  precisely,  Work = Force dot Displacement  .  You can get increased
  potential energy via the work you do against gravity (a force).

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12.4 Cross Product

The cross product is an idea valid in any dimension, but it is really
a number of ideas rolled into one.  These ideas work in any dimension,
but in 3-dimensions they can be viewed as a single "product."

FUNDAMENTAL IDEA: Much about the cross product follows from the properties 
  of 2 x 2 and 3 x 3 determinants (e.g., switching two rows results in 
  the same determinant with a - sign, multiplying any row by 42 will
  multiply the determinant by 42, etc.).  You may have seen this, but
  determinants are not typically covered in generality until you take a 
  course in linear algebra.

Cross products are best understood as arising from the 3 x 3 determinant, 
via det(a,b,c) = + or - the volume of the parallelopiped with side lengths
  a,b, and c.  We define the cross product via
    det(a,b,c) = a . CrossProd(b,c),  which gives the formula
CrossProd(b,c) = ( b_2 c_3 - b_3 c_2, etc., etc. ) .
The components of the cross product can be viewed as 2 x 2 determinants,
and so we conclude
  CrossProduct(b,c) = - CrossProduct(c,b)
  CrossProduct(b,7c) = 7 CrossProduct(b,c)
  CrossProduct(b,c+d) =  CrossProduct(b,c) + CrossProduct(b,d)
  CrossProduct(b,c) = 0 if b=c,
  etc.

Because 
    det(a,b,c) = a . CrossProd(b,c) 
and the fact that det(a,b,c) = 0 if a=b or a=c, we conclude

KEY PROPERTY:  a . CrossProd(b,c) = 0  if a=b or a=c, i.e., 
  CrossProd(b,c) is always orthogonal to b and c.

One can use parallelopiped volume property of the determinant to conclude
KEY PROPERTY 2:  if b and c are orthogonal unit vectors, then
  CrossProd(b,c) is also a unit vector.

Because 
    det(a,b,c) = a . CrossProd(b,c) ,
and the fact that
  det(a,b,c) = - det(b,a,c) =  - det(a,c,b) = det(c,a,b) = etc.
we conclude that
  a . CrossProd(b,c) = - b . CrossProd(a,c) = - a . CrossProd(c,b)
  = c . CrossProd(a,b) = etc.

Because of "physics cross product" property we see that
  | CrossProd(b,c) | = |b| |c| | sin(theta) |,
where theta is the angle between b and c.
  

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