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MAC 2313CALC III

Chapter 12

VECTORS and the GEOMETRY of SPACE

THOMAS’ CALCULUS – EARLY TRANSCENDENTALS, 11TH ED.

Commentary by Doug JonesRevised Aug. 14, 2010

§12.2

What’s a Vector?

Basic Definitions, Concepts and Ideas

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Section 12.2Vectors

• 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

• Can be Identified with the point

– 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?

1 2 1 2 1 2ˆ, , ,a a or a a a or a a a a

1 2,A a a

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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.

• This defines the Unit Vector u in the same direction

as a.

– What is the angle between the vector and the positive x-axis?

1, Given we definea u a

a

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1u 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.

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.”

a V b V a b V

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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:

“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.”

a V a V

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Addition Axioms• Commutative Property –

– Or (restated)

– where 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.”

a V b V a b b a

, , a b V a b b a

" " means "for each," "for every," or "for all,"

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• Associative Property of Vector Addition –

• Zero Vector – There exists an element of V, called the zero vector, and denoted by 0, which is such that

, , , a b c V a b c a b c

,

,

or

a V a 0 a

0 V, a V a 0 a

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• Existence of Negatives (Additive Inverse) –

(Instead of writing a' we usually write –a.)

Note: The symbol

and the symbol

" " " " or " " or " ."means there exists there is there are

, : a V a V a a 0

" :" " ".means such that

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Scalar Multiplication Axioms

•

, , , a b a b a b V

, , , a V a a a

, , , a V a a

, 1 a V a 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.

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

We could just as easily have displayed three dimensional vectors, four dimensional vectors, or even“n”-dimensional vectors.

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1 2,a a a

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1 2 3, ,a a aa

or

1 2 3, , , , na a a aa

1 2 3 4, , ,a a a aa

or

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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 are two 3-dimensional vectors, then the dot product is defined as

1 2 3 1 2 3, , and , ,a a a b b b a ba b

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1 1 2 2 3 31

def or

i ii

a b a b a b a b

a b

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For example, and we always give an example, if

12 5 and 1 4 9

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a , , b , ,

then

12 1 5 4 9 2 20 3 19

3a b

19a b

That is

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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

instead of a

.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

1 2 3, ,a a a a

is defined by

This is called the “Euclidean Norm.”

2 2 21 2 3a a a a

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Recall that

1 1 2 2 3 3a b a b a b a b

thus

2 2 21 1 2 2 3 3 1 2 3

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a a a a a a a a a a a

a

2a a a

So 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

θ

b

a

θ

b

c a b

Consider the two vectors and on the left. The included angle is θ. Well, now, we can insert the vector as shown on the right, thereby creating a triangle.

a

b

c

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And this triangle obeys the Law of Cosines, which in this case says that

2 2 22 cosc a b a b

Now, since the dot product follows the rules outlined above,

2

2 22

c c c a b a b

a b a a b b

a a b a a b b b a b a b

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So, in summary,2 2 2

2c a b a b

But, wait a minute! We already had that 2 2 2

2 cosc a b a b

Now, don’t these two facts mean that

cosa b a b

And this is the result that I was after!

?