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MAC 2313
CALC III
THOMAS’ CALCULUS – EARLY TRANSCENDENTALS, 11TH ED.
Chapter 12
VECTORS and the GEOMETRY of SPACE
Commentary by Doug Jones
Revised Aug. 14, 2010
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§12.2
What’s a Vector?
Basic Definitions, Concepts and
Ideas
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Section 12.2
Vectors
• You’ve studied vectors in Trig.
– Do you remember?
– What do you remember?
• Class Discussion…. (Somebody please
say something!)
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3 ways to understand vectors
1. Geometrically – ( by way of arrows )
2. Analytically – (by way of “ordered pairs” )
3. Axiomatically – (by way of axioms )
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Brief Review of #1 & #2
• #1. GEOMETRIC VECTORS
– Arrows → “Length” and “Direction”
• Discuss the differences between vectors & “old fashioned”
numbers (scalars).
– What can we do with arrows (vectors)?
• Ask questions
– but not “Are they green?” Or maybe they are!
• Add? Subtract? Multiply? Divide? How?
– Std. position
• Standard Position – what does this mean?
• Free vector – what does this mean?
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• #2. ANALYTIC VECTORS – remember
Analytic Geometry?
– Ordered Pairs.
• The Vector
a  a1 , a 2
or a  a1 , a 2
• Can be Identified with the point
or aˆ  a1 , a 2
A  a1 , a 2 
– But how does one add and subtract
analytically? (Discussion)
– What about the length of a vector?
– What about the angle that a vector makes
with the horizontal?
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– We can “size” a vector…. What’s this all about?
• The key is the normalized (or unit) vector.
– Normalizing a vector – Creating a Unit Vector.
– How to do it.
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Given a , we define u 
a
a
• This defines the Unit Vector u in the same direction
as a.
– What is the angle between the vector and the
positive x-axis?
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1
u
a
a
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OK, Here’s #3
Axiomatic Definition of Vectors
• A vector space V defined “over” the set of real
numbers R (called scalars) is a non-empty set V of
objects, called vectors, together with two operations,
scalar multiplication and vector addition, · and +
.
The system
•
•
V =(V, R, · , + )
satisfies the following axioms –
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Closure Axiom (1)
• The sum of two vectors is a vector.
a  V   b  V   a  b  V 
How does one pronounce this in English?
“If a is a vector and b is a vector, then a+b is a
vector.”
• Thus, a vector space V is said to be “closed
under vector addition.”
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Closure Axiom (2)
• The product of a scalar and a vector is a
vector. (Not a dog or a cat or a ham sandwich.)
In symbols:
    a  V   a  V 
“If α is a real number and a is a vector, then αa is a vector.”
• Thus, a vector space V is said to be “closed
under multiplication by a scalar.”
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Addition Axioms
• Commutative Property –
a  V   b  V   a  b  b  a 
– Or (restated)
 a, b  V, a  b  b  a
– where "  " means "for each," "for every," or "for all,"
whichever is appropriate for the context. Thus, the commutative
property, stated in words is:
“If a and b are vectors, then a+b=b+a.” or
“For any vectors, a and b, a+b=b+a.”
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• Associative Property of Vector Addition –
 a, b, c  V, a  b  c   a  b   c
• Zero Vector – There exists an element of V,
called the zero vector, and denoted by 0, which
is such that
 a  V, a  0  a
or
 0  V,  a  V , a  0  a
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• Existence of Negatives (Additive Inverse) –
 a  V,  a  V : a  a  0
(Instead of writing a' we usually write –a.)
Note: The symbol
" " means " there exists " or " there is " or " there are."
and the symbol
"  :" means " such that ".
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Scalar Multiplication Axioms
•
  ,  a, b V ,   a  b    a   b
•
 ,   , a  V,     a   a   a
•
 ,   , a  V,   a     a 
•
 a  V, 1a  a
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Conclusion
• When dealing with vectors, it is best if you
can consider the problem from all three
viewpoints (almost) simultaneously!
• Think of a vector as an arrow with an
ordered pair at its terminal point
(arrowhead) and which obeys certain
laws of combination and manipulation.
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Next Idea. . .
What’s a “Dot Product?”
§12.3
Another type of multiplication!
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A bit of an Introduction
• So far we have displayed two dimensional
vectors, such as
a  a1 , a2
We could just as easily have displayed
three dimensional vectors, four
dimensional vectors, or even
“n”-dimensional vectors.
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a  a1, a2 , a3
or
a  a1, a2 , a3 , a4
or
a  a1, a2 , a3 ,
, an
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However, in this course we’ll restrict
ourselves to two and three dimensional
vectors, and on occasion we’ll deal with a
four dimensional vector.
Note: If you are really interested in this
stuff, you can generalize this concept to
infinite dimensional vectors. As a matter
of fact, you have already studied what
amounts to one type of infinite
dimensional vector in Calc II under the
guise of infinite sequences.
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DOT PRODUCT
– SCALAR PRODUCT –
INNER PRODUCT
These are three different names for the same process.
We’ll usually call it the “dot product.”
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We define the dot product first analytically, and then we’ll
establish some rules of operation for the dot product.
Finally, we’ll derive the formulation of the dot product in
geometric terms. That is our plan.
We’ll do our work in terms of 3-dimensional vectors. Thus
our results will obviously cover the 2-dimensional case and
easily generalize to the n-dimensional case.
Definition: If a  a1 , a2 , a3 and b  b1 , b2 , b3 are
two 3-dimensional vectors, then the dot product a  b is
defined as
def
or
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a  b  a1b1  a2b2  a3b3   aibi
i 1
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For example, and we always give an example, if
1
a  2, 5 ,
3
and b  1, 4 ,9
then
1
a b  2  1   5  4  9  2  20  3  19
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That is
a b  19
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So, now can you see why they also call the
dot product the scalar product? It is because
the result of the dot product of two vectors is
not a vector, it is a real number – and a real
number is a number which can be located
on a ruler, i.e., on a scale.
Thus the dot product of two vectors is a
scalar; hence, the operation is often called
the scalar product in order to emphasize this
point.
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Also, please notice that in this last example
I’m using a different form of notation for
vectors. The letter with the arrow over it is
probably how most of you will be writing your
vectors, because it takes a bit of time and
effort to draw a boldfaced letter with pen or
pencil. Thus, we’ll usually write
a
instead of
a.
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Now we’ve got a few rules that we need to learn
in addition to the basic axioms of vectors,
discussed earlier:
The norm (or magnitude) of a vector
a  a1, a2 , a3
is defined by
a  a12  a2 2  a32
This is called the “Euclidean Norm.”
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Recall that
a  b  a1b1  a2b2  a3b3
thus
a  a  a1a1  a 2 a 2  a3a3  a1  a 2  a3
2
 a
So
2
2
2
aa  a
2
This will be useful to
us in the near future.
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Also
a b  ba

    
a  b  c  a b  a c
And the proofs of these are simply a matter
of verification.
Now let’s put these ideas to good work!
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a
a
c  a b
θ
θ
b
b
Consider the two vectors a and b on the left.
The included angle is θ. Well, now, we can
insert the vector
c
as shown on the right, thereby creating a
triangle.
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And this triangle obeys the Law of
Cosines, which in this case says that
2
2
2
c  a  b  2 a b cos 
Now, since the dot product follows the rules
outlined above,
 
 a  ba  a  bb
c
2

 cc  a b  a b
2
2
 a  a  b  a  a  b  b  b  a  b  2a  b
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2
So, in summary,
2
2
c  a  b  2a  b
But, wait a minute! We already had that
2
2
2
c  a  b  2 a b cos 
Now, don’t these two facts mean that
a  b  a b cos 
?
And this is the result that I was after!
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