Addition of Vectors

Imagine that you have a map that leads
you to a buried treasure.

This map has instructions such as 15
paces west northwest
of the skull.

The 15 paces is
a distance and west
northwest is a direction.
N


Quantities that are described by a
magnitude (such as the distance on the
previous slide) and a direction are called
vectors.
Those quantities that have no direction
are called scalars.


Examples of scalars in physics are
mass
time
distance
speed
work
energy
Examples of vectors in physics are
displacement
velocity
acceleration
force
momentum
angular momentum



The math associated with scalars is
familiar to everyone.
The math associated with vectors is
more involved.
Today you will explore the graphical
addition of vectors.


Let’s use a treasure map again as an
example of the addition of vectors.
Let’s imagine the instructions tell
you to go 4 miles east then 3 miles
north.
N
W
E
S
5 miles
3 miles
36.90
4 miles



In this case you could have gone 3
miles north first and then 4 miles
east next and still end up at the same
location.
Your final position is 5 miles at 36.90
north of east.
It would have saved time if that had
been the one distance and one
direction traveled in the first place.



We say that the 5 miles at 36.90 north
of east is the vector sum of the 4 miles
east vector and the 3 miles north
vector.
The order of the addition does not
matter.
Examples of addition of vectors follows.
The method used is the
head-to-tail method.

You must know how to do these for your
lab exams.
Head-to-Tail Method of Addition of Two Vectors
Head-to-Tail Method of Addition of Three Vectors
You will need to use a scale when you
graph your vectors.
For example, you might set 10 cm to
represent 100 grams.
How to Construct
Two Perpendicular Lines
(With a Protractor)
to form a
Set of Coordinate Axes
900
We will consider the
following three
force vectors acting
at the origin of this
coordinate system.
00
1800

F1  3 N @ 45 

F2  1 N @ 60 

F3  2 N @ 210 
2700
900
We will add these
force vectors by the
head-to-tail method.
We will see how this
is done
 over the next
few F
slides.
 3 N @ 45 
1
00
1800
2700
How to Construct
a Second Set of
Coordinate Axes Parallel to
an Initial Set of
Coordinate Axes
Mark the angle
for the vector.
Remember the
angle marked.

F3  2 N @ 210 
900
Resultant
00
0
180

F1  3 N @ 45 

F2  1 N @ 60 
   
R  F1  F2  F3

R  2.18 N @ 65.9 
2700
900

F1
900
Resultant
0
180

1800

F2
00
F1  3 N @ 45
F

F2  1 N @ 60 

F3  2 N @ 210 
3
00
 Equilibrant
  
R  F1  F2  F3

R  2.18 N @ 65.9 
2700
2700
Note that the equilibrant has the same
magnitude as the resultant but is in
the opposite direction.
Therefore it “balances” out the
resultant which also means it
“balances” out the three forces, too.