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
Adding Vectors, Rules final


When adding vectors, all of the vectors must
have the same units
All of the vectors must be of the same type of
quantity

For example, you cannot add a displacement to a
velocity
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

 
A  A  0
The negative of the vector will have the same
magnitude, but point in the opposite direction


Represented as A
Subtracting Vectors



Special case of vector
addition
If A  B , then use A  B
Continue with standard
vector addition
procedure
 
Subtracting Vectors, Method 2

Another way to look at
subtraction is to find the
vector that, added to
the second vector gives
you the first vector

A  B  C
 

As shown, the resultant
vector points from the tip
of the second to the tip of
the first
Multiplying or Dividing a
Vector by a Scalar




The result of the multiplication or division of a vector
by a scalar 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
Component Method of Adding
Vectors

Graphical addition is not recommended when



High accuracy is required
If you have a three-dimensional problem
Component method is an alternative method

It uses projections of vectors along coordinate
axes
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
Vector Component
Terminology

A x and A y are the component vectors of A


They are vectors and follow all the rules for
vectors
Ax and Ay are scalars, and will be referred to
as the components of A
Components of a Vector




Assume you are given
a vector A
It can be expressed in
terms of two other
vectors, A x and A y
These three vectors
form a right triangle
A  Ax  Ay
Components of a Vector, 2


The y-component is
moved to the end of
the x-component
This is due to the fact
that any vector can be
moved parallel to
itself without being
affected

This completes the
triangle
Components of a Vector, 3

The x-component of a vector is the projection along
the x-axis
Ax  A cos 

The y-component of a vector is the projection along
the y-axis
Ay  A sin 

This assumes the angle θ is measured with respect
to the x-axis

If not, do not use these equations, use the sides of the
triangle directly
Components of a Vector, 4

The components are the legs of the right triangle
whose hypotenuse is the length of A
2
2
1 Ay
A  Ax  Ay and   tan
Ax

May still have to find θ with respect to the positive x-axis
Components of a Vector, final


The components can
be positive or negative
and will have the same
units as the original
vector
The signs of the
components will
depend on the angle
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
Unit Vectors, cont.

The symbols
î , ĵ, and k̂


represent unit vectors
They form a set of
mutually perpendicular
vectors in a righthanded coordinate
system
Remember, ˆi  ˆj  kˆ  1
Viewing a Vector and Its
Projections


Rotate the axes for
various views
Study the projection of
a vector on various
planes



x, y
x, z
y, z
Unit Vectors in Vector Notation


Ax is the same as Ax î
and Ay is the same as
Ay ĵ etc.
The complete vector
can be expressed as
A  Ax ˆi  Ay ˆj
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
  tan
1
Ry
Rx
Adding Vectors with Unit
Vectors



Note the relationships
among the components
of the resultant and the
components of the
original vectors
Rx = Ax + Bx
R y = A y + By
Three-Dimensional Extension


Using R  A  B
Then R  Ax ˆi  Ay ˆj  Azkˆ  Bx ˆi  By ˆj  Bzkˆ

 

R   Ax  Bx  ˆi   Ay  By  ˆj   Az  Bz  kˆ
R  Rx ˆi  Ry ˆj  Rzkˆ

and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Ax+Bz
R  R R R
2
x
2
y
2
z
Rx
  cos
, etc.
R
1
Example 3.5 – Taking a Hike

A hiker begins a trip by first walking 25.0 km
southeast from her car. She stops and sets up her
tent for the night. On the second day, she walks
40.0 km in a direction 60.0° north of east, at which
point she discovers a forest ranger’s tower.
Example 3.5

(A) Determine the components
of the hiker’s displacement for
each day.
A
Solution: We conceptualize the problem by drawing a
sketch as in the figure above. If we denote the
displacement vectors on the first and second days by A
and B respectively, and use the car as the origin of
coordinates, we obtain the vectors shown in the figure.
Drawing the resultant R , we can now categorize this
problem as an addition of two vectors.
Example 3.5

We will analyze this
problem by using our new
knowledge of vector
components. Displacement A
has a magnitude of 25.0 km
and is directed 45.0° below
the positive x axis.
From Equations 3.8 and 3.9, its components are:
Ax  A cos( 45.0)  (25.0 km)(0.707) = 17.7 km
Ay  A sin( 45.0)  (25.0 km)( 0.707)  17.7 km
The negative value of Ay indicates that the hiker walks in the
negative y direction on the first day. The signs of Ax and Ay
also are evident from the figure above.
Example 3.5

The second
displacement B has a
magnitude of 40.0 km
and is 60.0° north of
east.
Its components are:
Bx  B cos60.0  (40.0 km)(0.500) = 20.0 km
By  B sin 60.0  (40.0 km)(0.866)  34.6 km
Example 3.5

(B) Determine the
components of the hiker’s
resultant displacement R
for the trip. Find an
R
expression
for in terms of
unit vectors.
R
Solution: The resultant displacement for the trip R  A  B
has components given by Equation 3.15:
Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km
Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km
In unit-vector form, we can write the total displacement as
R = (37.7 ˆi + 16.9ˆj) km
Example 3.5

Using Equations 3.16 and
3.17, we find that the
resultant vector has a
magnitude of 41.3 km and
is directed 24.1° north of
east.
R
Let us finalize. The units of R are km, which is reasonable for a
displacement. Looking at the graphical representation in the
figure above, we estimate that the final position of the hiker is at
about (38 km, 17 km) which is consistent with the components
of R in our final result. Also, both components of R are positive,
putting the final position in the first quadrant of the coordinate
system, which is also consistent with the figure.
Motion in Two Dimensions

Using + or – signs is not always sufficient to fully
describe motion in more than one dimension




Vectors can be used to more fully describe motion
Will look at vector nature of quantities in more detail
Still interested in displacement, velocity, and
acceleration
Will serve as the basis of multiple types of motion in
future chapters
Position and Displacement

The position of an
object is described by
its position vector, r
The displacement of
the object is defined as
the change in its
position

r  rf  ri

General Motion Ideas

In two- or three-dimensional kinematics,
everything is the same as as in onedimensional motion except that we must now
use full vector notation

Positive and negative signs are no longer
sufficient to determine the direction
Average Velocity



The average velocity is the ratio of the displacement
to the time interval for the displacement
r
vavg 
t
The direction of the average velocity is the direction
of the displacement vector
The average velocity between points is independent
of the path taken

This is because it is dependent on the displacement, also
independent of the path
Instantaneous Velocity

The instantaneous
velocity is the limit of the
average velocity as Δt
approaches zero
r dr
v  lim

dt
t 0 t

As the time interval
becomes smaller, the
direction of the
displacement approaches
that of the line tangent to
the curve
Instantaneous Velocity, cont


The direction of the instantaneous velocity
vector at any point in a particle’s path is along
a line tangent to the path at that point and in
the direction of motion
The magnitude of the instantaneous velocity
vector is the speed

The speed is a scalar quantity
Average Acceleration

The average acceleration of a particle as it
moves is defined as the change in the
instantaneous velocity vector divided by the
time interval during which that change
occurs.
aavg
v f  vi v


tf  t i
t
Average Acceleration, cont

As a particle moves,
the direction of the
change in velocity is
found by vector
subtraction
v  vf  vi

The average
acceleration is a vector
quantity directed along
v
Instantaneous Acceleration

The instantaneous acceleration is the limiting
value of the ratio v t as Δt approaches
zero
v dv
a  lim

dt
t 0 t

The instantaneous equals the derivative of the
velocity vector with respect to time
Producing An Acceleration

Various changes in a particle’s motion may
produce an acceleration


The magnitude of the velocity vector may change
The direction of the velocity vector may change


Even if the magnitude remains constant
Both may change simultaneously
Kinematic Equations for TwoDimensional Motion



When the two-dimensional motion has a constant
acceleration, a series of equations can be
developed that describe the motion
These equations will be similar to those of onedimensional kinematics
Motion in two dimensions can be modeled as two
independent motions in each of the two
perpendicular directions associated with the x and y
axes

Any influence in the y direction does not affect the motion
in the x direction
Kinematic Equations, 2


Position vector for a particle moving in the xy
plane r  x ˆi  yˆj
The velocity vector can be found from the
position vector
dr
v
 v x ˆi  v y ˆj
dt

Since acceleration is constant, we can also find
an expression for the velocity as a function of
time: vf  vi  at
Kinematic Equations, 3

The position vector can also be expressed as
a function of time:


rf  ri  vi t  1 at 2
2
This indicates that the position vector is the sum
of three other vectors:



The initial position vector
The displacement resulting from the initial velocity
The displacement resulting from the acceleration
Kinematic Equations, Graphical
Representation of Final Velocity


The velocity vector can
be represented by its
components
v f is generally not along
the direction of either vi
or a
Kinematic Equations, Graphical
Representation of Final Position



The vector
representation of the
position vector
rf is generally not along
the same direction as vi
or as a
v f and rf are generally
not in the same
direction
Graphical Representation
Summary


Various starting positions and initial velocities
can be chosen
Note the relationships between changes
made in either the position or velocity and the
resulting effect on the other