Chapter 3
Lecture 4:
Vectors
HW2 (problems): 2.70, 2.72, 2.78,
3.5, 3.13, 3.28, 3.34, 3.40
Due next Friday, Feb. 13
Kinematic Equations from
Calculus

Displacement equals
the area under the
velocity – time curve
lim
 tn  0

v
n
tf
xn
tn   v x (t )dt
ti
The limit of the sum is a
definite integral
2-10: Graphical integration in motion analysis
Starting from
we obtain
(vo= velocity at time t=0, and v1=
velocity at time t = t1).
Note that
Similarly, we obtain
(xo= position at time t = 0, and x1 =
position at time t=t1), and
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
Cartesian Coordinate System



Also called rectangular
coordinate system
x- and y- axes intersect
at the origin
Points are labeled (x,y)
Polar to Cartesian Coordinates



Based on forming
a right triangle
from r and q
x = r cos q
y = r sin q
Cartesian to Polar Coordinates

r is the hypotenuse and q
an angle
tanq 
y
x
r  x2  y 2

q must be ccw from positive
x axis for these equations to
be valid
Adding Vectors Graphically



Continue drawing the
vectors “tip-to-tail”
The resultant is drawn
from the origin of A to
the end of the last
vector
Measure the length of R
and its angle

Use the scale factor to
convert length to actual
magnitude
Adding Vectors Graphically,
final


When you have many
vectors, just keep
repeating the process
until all are included
The resultant is still
drawn from the tail of
the first vector to the tip
of the last vector
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
Adding Vectors, Rules cont.

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
Negative of a Vector

The negative of a vector is defined as the
vector that, when added to the original vector,
gives a resultant of zero
 Represented as A
 A  A  0
The negative of the vector will have the same
magnitude, but point in the opposite direction
 

Subtracting Vectors



Special case of vector
addition
If A  B , then use A  B
Continue with standard
vector addition
procedure
 
Components of a Vector,
Introduction

A component is a
projection of a vector
along an axis


Any vector can be
completely described by
its components
It is useful to use
rectangular
components

These are the projections
of the vector along the xand y-axes
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
The symbols
ˆi ,ˆj, and kˆ
represent unit vectors
They form a set of mutually
perpendicular vectors in a righthanded coordinate system
Remember, ˆi  ˆj  kˆ  1
Adding Vectors Using Unit
Vectors


Using R  A  B
Then R  Ax ˆi  Ay ˆj  Bx ˆi  By ˆj

 

R   Ax  Bx  ˆi   Ay  By  ˆj
R  Rx ˆi  Ry ˆj

and so Rx = Ax + Bx and Ry = Ay + By
R  R R
2
x
2
y
q  tan
1
Ry
Rx
3.8: Multiplying vectors
A. Multiplying a vector by a scalar
Multiplying a vector by a scalar
changes the magnitude but not the
direction:


a x s  sa
3.8: Multiplying vectors
B. Multiplying a vector by a vector: Scalar (Dot)
Product
The scalar product between
two vectors is written as:
It is defined as:
Here, a and b are the
magnitudes of vectors a and
b respectively, and f is the
angle between the two
vectors. The right hand side
is a scalar quantity.
3.8: Multiplying vectors
C. Multiplying a vector with a vector: Vector (Cross)
Product
The vector product between
two vectors a and b can be
The right-hand rule allows us to find
the direction of vector c.
written as:
The result is a new vector c, which
is:
Here a and b are the
magnitudes of vectors a and b
respectively, and f is the
smaller of the two angles
between a and b vectors.
3.8: Multiplying vectors; vector product in unit-vector notation:
Note that:
ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  0
ˆi  ˆj  ˆj  ˆi  kˆ
ˆj  kˆ  kˆ  ˆj  ˆi
kˆ  ˆi  ˆi  kˆ  ˆj
Sample problem, vector product, unit vector notation