Download Dot Product

Goals of Lecture 2
• Introduce Interactive Learning Segments and try a
few
• The Language of Vectors:
– Understand conventions used in denoting vectors.
– Visualize vector operations both from the geometric and
algebraic point of view.
• Dot Products:
– Understand geometric and algebraic formulation of the
vector dot product.
– Understand some properties of the dot product
Phy 221 2005S Lecture 2
What is a Vector
• A scalar quantity is one that is represented by a
single number (e.g. Mass Length time temperature
volume…)
• A vector is a quantity which has both magnitude and
direction (e.g. displacement, velocity, force)
• Magnitude: How long is the vector
• Direction: angle counterclockwise from x-axis or
other sensible description.
• Geometrically: we represent a vector as an arrow
Phy 221 2005S Lecture 2
Basics
• Equal Vectors:
vectors are equal if they have the same
magnitude and direction regardless of where the vector
“starts”

A
 
AB

B
• Opposite Vectors:
Vector are opposite if the magnitude
is the same but direction is opposite

A


A  B

• Unit Vectors: Â is in the direction of Abut of length 1.

B
Vector Addition
A+B=C
Geometrically: Parallel transport the tail of B to the head
of A. The sum goes from the tail of A to the head of B.
Algebraically: Add the components
C(3,5)
B(-2,4)
A (5,1)
A 5
B -2
C 3
1
4
5
Vector addition is commutative and associative
Phy 221 2005S Lecture 2
interACTive learning segments (ACTs).
• Get your clickers ready
• I will post a question, you should talk to your
neighbor and “vote” on the correct answer
with your clicker
• You can change your answer if you change
your mind.
• When you click, a box with your clicker
number will appear
• At the end of the time, a bar graph will show
the summary of results.
Phy 221 2005S Lecture 2
ACT: Vector addition
•
All the vectors below have the same magnitude. Which of the
following arrangements will produce the largest resultant when
the two vectors are added?
1.
2.
Phy 221 2005S Lecture 2
3.
The language of vectors
• To keep straight which variables are scalar
quantities, it is conventional to draw a little
arrow over vector variables. A B C
• If A is a vector, we use A (without arrow) to
denote the magnitude.
• Sometimes boldface is used for vector and
normal font for magnitude. Thus vector=A;
magnitude=A
• More formally, we can indicate magnitude of
a vector by vertical bars.
• Thus A=|A|.
Phy 221 2005S Lecture 2
Vector Addition
• Geometric:
B
C
A+B=C
A
• Algebraic: Ax+Bx=Cx
Ay+By=Cy
Az+Bz=Cz
• Subtraction:
D
B
A-B=D
Note two ways to think:
A
-B D is A plus –B
D goes from tip of B to tip of A
when A and B are rooted at a
Phy 221 2005S Lecture 2 common point
Components
• The components of a vector can be thought of as the
projections along the coordinate axes. These are sometimes
called the Cartesian coordinates:
• We can denote A in terms of its components as: A=(Ax,Ay,Az)
y
Ay
A
x
ALecture
Phy 221 2005S
2
x
Unit Vectors
• A unit vector is a vector of length 1. To indicate a vector is a
unit vector, we put a hat on it:
Â
denotes a unit vector parallel to

A
• Some special unit vectors
– The unit vector that points along the x axis is denoted ^
i
– The unit vector that points along the y axis is denoted j^
– The unit vector that points along the z axis is denoted ^
k
y
Any vector can be written in terms of these basic unit vectors
If A=(Ax,Ay,Az) then
^
j
A
x
y
z
^
i
x
^k

A  A iˆ  A ˆj  A kˆ
z
Phy 221 2005S Lecture 2
Polar Notation
• In 2 dimensions, one can also describe a
vector by its magnitude and direction.
• The direction is the angle  taken
counterclockwise from the x axis
| A | Ax  Ay
 Ay 
Ax | A | cos 
  arctan  
Ay | A | sin 
 Ax 
 Ax 

How do I remember which is  arccos 
 A 
sin and which is cos?
2
A
|A|

Phy 221 2005S Lecture 2
2
Dot Products
• Dot Products are the workhorse of vector
analysis
Definition
Algebraic
Geometric
•In
 terms of components:
•If  is the angle between A and B:
A  B  Ax Bx  Ay By  Az Bz
 
A  B | A || B | cos 
Where does
this come from?
B

A
Phy 221 2005S Lecture 2
Properties of The Dot Product
• The dot product takes two vectors as inputs and produces a
scalar as output.
• The dot product is commutative: A•B=B•A
  
   
• Dot product distributes over vector addition: A  ( B  C )  A  B  A  C

• The dot product between a vector, A, and a unit, u, vector gives

the projection of A along u.
A
• In particular, the components
of A are the dot products with

u




i, j and
A•u
 k



Ax  A  i

Ay  A  j
Az  A  kˆ
• The length of a vector can be expressed in terms of the dot
product: |A|²=A•A
Phy 221 2005S Lecture 2
Some Special Cases
• Vectors that are going in the same direction
 
A  B  AB
• Vectors that are going in opposite directions
 
A  B   AB
• Perpendicular vectors
 
A B  0
Remember
vector #1
Note: Later in the course we will learn
about another kind of product called
the “cross product” which takes in two
vectors and spits out another vector. Do
not get them confused.
Phy 221 2005S Lecture 2
vector #2
Dot Product
Scalar
Angle Between Vectors
• We can use the geometric definition of the dot
product to determine the angle between two
vectors:
 
   
A B
A  B | A || B | cos 
cos   
| A || B |
• This tells the angle between A and B but not the
direction of the angle
B

A
Phy 221 2005S Lecture 2
Example of Vector Algebra:
Law of cosines
Consider a triangle, The sum of the vectors representing the sides is 0
C
B

p
A
C=-(A+B)
C²=(A+B)²=(A+B)·(A+B)
=A²+B²+2A·B
=A²+B²+2AB cos(p)
=A²+B²-2AB cos()
Phy 221 2005S Lecture 2
Binomial expansion of
dot product: a useful
trick-learn it!