Directions – Pointed
Things
January 23, 2005
January 23, 2006
Vectors
1
Today …
• We complete the topic of kinematics in one
dimension with one last simple (??) problem.
• We start the topic of vectors.
– You already read all about them … right???
• There is a WebAssign problem set assigned.
• The exam date & content may be changed.
– Decision on Wednesday
– Study for it anyway. “may” and “will” have different
meanings!
January 23, 2006
Vectors
2
One more Kinematics Problem
A freely falling object requires 1.50 s to travel the
last 30.0 m before it hits the ground. From what
height above the ground did it fall?
h=??
30 meters
1.5 seconds
January 23, 2006
Vectors
3
The Consequences of Vectors



You need to recall your trigonometry if you
have forgotten it.
We will be using sine, cosine, tangent, etc.
thingys.
You therefore will need calculators for all
quizzes from this point on.


January 23, 2006
A cheap math calculator is sufficient,
Mine cost $11.00 and works just fine for most of my
calculator needs.
Vectors
4
Hey … I’m lost. How do I get to lake
Eola
B
NO PROBLEM!
Just drive 1000 meters and
you will be there.
HEY!
Where the
*&@$ am I??
A
January 23, 2006
Vectors
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The problem:
January 23, 2006
Vectors
6
IS this the way (with direecton now!) to
Lake Eola?
B
THIS IS A VECTOR
A
January 23, 2006
Vectors
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YUM!!
January 23, 2006
Vectors
8
The following slides were
supplied by the publisher …
…………… thanks guys!
January 23, 2006
Vectors
9
Vectors and Scalars


A scalar quantity is completely specified by
a single value with an appropriate unit and
has no direction.
A vector quantity is completely described by
a number and appropriate units plus a
direction.
January 23, 2006
Vectors
10
Vector Notation






A
When handwritten, use an arrow:
When printed, will be in bold print: A
When dealing with just the magnitude of a vector in
print, an italic letter will be used: A or |A|
The magnitude of the vector has physical units
The magnitude of a vector is always a positive
number
January 23, 2006
Vectors
11
Vector Example

A particle travels from A to B
along the path shown by the
dotted red line


This is the distance
traveled and is a scalar
The displacement is the
solid line from A to B


The displacement is
independent of the path
taken between the two
points
Displacement is a vector
January 23, 2006
Vectors
12
Equality of Two Vectors



Two vectors are equal
if they have the same
magnitude and the
same direction
A = B if A = B and they
point along parallel
lines
All of the vectors shown
are equal
January 23, 2006
Vectors
13
Coordinate Systems


Used to describe the position of a point in
space
Coordinate system consists of



a fixed reference point called the origin
specific axes with scales and labels
instructions on how to label a point relative to the
origin and the axes
January 23, 2006
Vectors
14
Cartesian Coordinate System



Also called rectangular
coordinate system
x- and y- axes intersect
at the origin
Points are labeled (x,y)
January 23, 2006
Vectors
15
Polar Coordinate System



Origin and reference line
are noted
Point is distance r from
the origin in the direction
of angle , ccw from
reference line
Points are labeled (r,)
January 23, 2006
Vectors
16
Polar to Cartesian Coordinates



Based on forming
a right triangle
from r and 
x = r cos 
y = r sin 
January 23, 2006
Vectors
17
Cartesian to Polar Coordinates

r is the hypotenuse and 
an angle
y
tan  
x
r  x2  y2

 must be ccw from positive
x axis for these equations to
be valid
January 23, 2006
Vectors
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Example

The Cartesian coordinates of a
point in the xy plane are (x,y) = (3.50, -2.50) m, as shown in the
figure. Find the polar coordinates
of this point.

Solution: From Equation 3.4,
y 2.50 m
tan   
 0.714
x 3.50 m
  216
and from Equation 3.3,
r  x 2  y 2  ( 3.50 m)2  ( 2.50 m)2  4.30 m
January 23, 2006
Vectors
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If the rectangular coordinates of
a point are given by (2, y) and its
polar coordinates are (r, 30),
determine y and r.
January 23, 2006
Vectors
20
Adding Vectors Graphically
January 23, 2006
Vectors
21
Adding Vectors Graphically
January 23, 2006
Vectors
22
Adding Vectors, Rules

When two vectors are
added, the sum is
independent of the
order of the addition.


This is the commutative
law of addition
A+B=B+A
January 23, 2006
Vectors
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Adding Vectors

When adding three or more vectors, their sum is independent
of the way in which the individual vectors are grouped


This is called the Associative Property of Addition
(A + B) + C = A + (B + C)
January 23, 2006
Vectors
24
Vector A has a magnitude of 8.00 units
and makes an angle of 45.0° with the
positive x axis. Vector B also has a
magnitude of 8.00 units and is directed
along the negative x axis. Using
graphical methods, find (a) the vector
sum A + B and (b) the vector
difference A  B.
January 23, 2006
Vectors
25
Subtracting Vectors



Special case of vector
addition
If A – B, then use A+(B)
Continue with standard
vector addition
procedure
January 23, 2006
Vectors
26
Multiplying or Dividing a Vector by a
Scalar




The result of the multiplication or division is a vector
The magnitude of the vector is multiplied or divided
by the scalar
If the scalar is positive, the direction of the result is
the same as of the original vector
If the scalar is negative, the direction of the result is
opposite that of the original vector
January 23, 2006
Vectors
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Each of the displacement vectors A
and B shown in Fig. P3.15 has a
magnitude of 3.00 m. Find
graphically (a) A + B, (b)
A  B, (c) B  A, (d) A  2B.
Report all angles counterclockwise
from the positive x axis.
January 23, 2006
Vectors
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Components of a Vector


A component is a part
It is useful to use
rectangular
components

These are the projections
of the vector along the xand y-axes
January 23, 2006
Vectors
29
Unit Vectors


A unit vector is a dimensionless vector with
a magnitude of exactly 1.
Unit vectors are used to specify a direction
and have no other physical significance
January 23, 2006
Vectors
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Unit Vectors, cont.

The symbols
î , ĵ, and k̂
represent unit vectors
 They form a set of
mutually perpendicular
vectors
January 23, 2006
Vectors
31
Unit Vectors in Vector Notation


Ax is the same as Ax
and Ay is the same as
Ay etc.
The complete
ĵ vector
can be expressed as
î
A  Ax ˆi  Ay ˆj  Az kˆ
January 23, 2006
Vectors
32
Adding Vectors with Unit Vectors
January 23, 2006
Vectors
33
Adding Vectors Using Unit Vectors –
Three Directions

Using R = A + B

 
R  Ax ˆi  Ay ˆj  Az kˆ  Bx ˆi  B y ˆj  Bz kˆ

R   Ax  Bx  ˆi   Ay  B y  ˆj   Az  Bz  kˆ
R  Rx  R y  Rz

Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz
etc.
R R R R
2
x
January 23, 2006
2
y
2
z
Vectors
Rx
 x  tan
R
1
34
Consider the two vectors and . Calculate (a) A
+ B, (b) A  B, (c) |A + B|, (d) |A  B|, and
(e) the directions of A + B and A  B .
January 23, 2006
Vectors
35
Three displacement vectors of a croquet ball are shown in the
Figure, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0
units. Find (a) the resultant in unit-vector notation and (b) the
magnitude and direction of the resultant displacement.
January 23, 2006
Vectors
36
The helicopter view in Fig.
P3.35 shows two people
pulling on a stubborn mule.
Find (a) the single force
that is equivalent to the two
forces shown, and (b) the
force that a third person
would have to exert on the
mule to make the resultant
force equal to zero. The
forces are measured in units
of newtons (abbreviated
N).
January 23, 2006
Vectors
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