Transcript Fun with Vectors

Fun with Vectors
Definition
• A vector is a quantity that has both
magnitude and direction
• Examples?
Represented by an arrow
B (terminal point)
v, v, or AB
A (initial point)
If two vectors, u and v, have the
same length and direction,
we say they are equivalent
v
u
Vector addition
a
b
Vector addition: a + b
a
b
Vector addition: a + b
b
a+b
a
Vector addition: a + b
b
a+b
a
Scalar Multiplication
a
Scalar Multiplication
a
2a
-a
-½ a
Subtraction
b
a
Subtraction
b
a
-b
Subtraction
b
a
a+(-b)
-b
Subtraction
b
a
a+(-b)
-b
Subtraction If a and b share the same initial
point, the vector a-b is the vector
from the terminal point of b
to the terminal point of a
b
a+(-b)
a
-b
Let’s put these on a coordinate
system
We can describe a vector by putting its initial
point at the origin.
We denote this as a=<a1,a2>
where (a1,a2) represent the terminal point
Graphically
z
y
(a,b,c)
(a1,a2)
c
y
a
x
a=<a1,a2>
b
x
v=<a,b,c>
Given two points A=(x1,y1) and B=(x2,y2),
The vector v = AB is given by
v = <x2 - x1, y2 - y1>
…or in 3-space,
v = <x2 - x1, y2 - y1, z2 - z1>
Graphically
A=(-1,2)
B=(2,3)
B
A
v = <2-(-1), 3-2> = <3,1>
v
Recall, a vector has direction and length
Definition: The magnitude of a vector
v = <x,y,z> is given by

2
2
2
v  x y z
Properties of Vectors
Suppose a, band c are vectors, c and d are scalars
1.
2.
3.
4.
5.
6.
7.
8.
a+b=b+a
a+(b+c)=(a+b)+c
a+0=a
a+(-a)=0
c(a+b)=ca+cb
(c+d)a=ca+da
(cd)a=c(da)
1a=a
Standard Basis Vectors

i  1,0,0

j  0,1,0

k  0,0,1
  
i  j  k 1
Definition: vectors with length 1 are called
unit vectors
Example: We can express
vectors in terms of this basis
a = <2,-4,6>
a = 2i -4j+6k
Q. How do we find a unit vector in the same
direction as a?
 1 
v

a


A. Scale a by its magnitude
a


1  a
v   a   1
a
a
Example
a = <2,-4,6>
 1 
v a
a

1
2  ( 4 )  6
2
2
1

2,4,6
56
2
2,4,6
12.3 The Dot Product
Motivation: Work = Force* Distance
F

Fx
Box
Fy
D
To find the work done in moving the box,
we want the part of F in the direction of the distance


Fx  F cos( )
 
 W  D F cos( )
F
Fy

Fx
Box
D
One interpretation of the dot
product
   
F  D  F D cos( )
Where
 is the angle between F and D
A more useful definition
 
a  b  a1b1  a2b2  a3b3
You can show these two definitions
are equal by considering the following triangle
and applying the law of cosines! See page 808 for details
 2 2 2
 
a  b  a  b  2 a b cos( )
z
a-b
b
Think, what is
a
|a|2?
y
x
Example
a=<2,-1,0>, b=<1,-8,-3>
Find a.b and the angle between a and b
The Dot Product
If a = <a1,a2,a3> and b=<b1,b2,b3> then
The dot product of a and b is a NUMB3R given
by
a b  a1b1  a2b2  a3b3
a  b  a b cos( )
The Dot Product
a b
cos( ) 
ab
a
cos( / 2)  0
b
a and b are orthogonal
if and only if the dot product of a and b is 0
Other Remarks:

0  

2
 cos( )  0
2
     cos( )  0
Properties of the dot product
Suppose a, b, and c are vectors and c is a scalar
1.
2.
3.
4.
5.
a.a=|a|2
a.b=b.a
a.(b+c) = (a.b)+(a.c)
(ca).b=c(a.b)=a.(cb)
0.a=0
Yet another use of the dot product:
Projections
a.b=|a| |b| cos()
Think of our work example:
this is ‘how much’ of b is in
the direction of a
b

|b| cos()
a
We call this quantity the
scalar projection of b on a


 a b 
compab    b cos( )
a
Think of it this way:
The scalar projection is the length of the shadow
of b cast upon a by a light directly above a
Q. How do we get the vector in the
direction of a with length compab?
A.We need to multiply the unit
vector in the direction of a by compab.
We call this the vector projection of b onto a


 a b
proja b  
a
  
a a b 
  2 a
a
a
Examples/Practice!
Key Points
• Vector algebra: addition, subtraction,
scalar multiplication
• Geometric interpretation
• Unit vectors
• The dot product and the angle between
vectors
• Projections (algebraic and geometric)