STATICS: CE201
Chapter 2
Force Vectors
Notes are prepared based on: Engineering Mechanics, Statics by R. C. Hibbeler, 12E Pearson
Dr M. Touahmia & Dr M. Boukendakdji
Civil Engineering Department, University of Hail
(2012/2013)
2.
Force Vectors
________________________________________________________________________________________________________________________________________________
Main Goals: In this chapter we define scalars, vectors and
vector operations and use them to analyze forces acting on
objects.
Content:
1. Scalars and Vectors
2. Vector Operations
3. Vector Addition of Forces
4. Coplanar Forces
5. Cartesian Vectors
6. Position Vectors
7. Dot Product
Chapter2 - Force Vectors
Scalars and Vectors

All physical quantities in engineering mechanics
are measured using either scalars or vectors.

Scalar: is any positive or negative physical
quantity that can be completely specified by its
magnitude (Ex: mass, time, length).

Vector: is any physical quantity that requires
both a magnitude and a direction for its
complete description (Ex: force, moment).
Chapter2 - Force Vectors
Vector

A vector is represented graphically by an arrow. The
length of the arrow represents the magnitude of the
vector and the angle θ between the vector and a fixed
axis defines the direction of its line of action. The head
of the arrow indicates the direction of the vector.

Vector quantities are represented either by a bold face
letters such as A and its magnitude
 is italicized A, or by
a character with an arrow on it: A
Chapter2 - Force Vectors
Vector Operations
Multiplication and Division of a Vector by a Scalar

The product of a vector A and a scalar a is a vector aA
with magnitude |aA | = |a | |A |

The direction of aA is the same as that of A if a is positive
and opposite to that of A if a is negative.
Chapter2 - Force Vectors
Vector Addition: The Parallelogram Rule
Two vectors A and B can be added to form a resultant vector
(equivalent vector) R = A + B using the parallelogram law .
This law states that: if A and B are two free vectors drawn
on scale, the resultant of the these vectors can be found by
drawing a parallelogram having sides of these vectors, and
the resultant will be the diagonal starting from the tails of
both vectors and ending at the heads of both vectors.
Chapter2 - Force Vectors
Vector Addition: The Triangle Rule

Using the triangle rule, vector B is added to vector A in
a “head-to-tail” fashion, by connecting the head of A to
the tail of B. The resultant R extends from the tail of A
to the head of B.

R can also be obtained by adding A to B:
R=A+B=B+A
Chapter2 - Force Vectors
Vector Addition: Collinear Vectors
If the two vectors A and B are collinear, i.e. both have the
same line of action, the parallelogram law reduces to an
algebraic or scalar addition: R = A + B
Chapter2 - Force Vectors
Vector Subtraction
The resultant of the difference between two vectors A and B
of the same type can be expressed as:
R’ = A – B = A + (– B)
Subtraction is defined as a special case of addition, so the
rules of vector addition also apply to vector subtraction.
Chapter2 - Force Vectors
Vector Addition of Forces

A force is a vector quantity since it has a specified
magnitude and direction.

Forces are added together or resolved into components
using the rules of vector algebra.

Two common problems in statics involve either finding
the resultant force given its components or resolving a
known force into components.
Chapter2 - Force Vectors
Finding a Resultant Force


Two “component” forces F1 and F2 added together,
according to the parallelogram law, yielding a resultant
force FR that form the diagonal of the parallelogram.
Examples:
Chapter2 - Force Vectors
Finding the Components of a Force
“Resolution” of a vector is breaking up a vector into
Components:
If a force F is to be resolved into components along two axes
u and v, then start at the head of force F and construct lines
parallel to the axes, thereby forming the parallelogram. The
sides of the parallelogram represent the components Fu and
Fv. This parallelogram can be reduced to a triangle.
Chapter2 - Force Vectors
Addition of Several Forces
If more than two forces are to be added, successive
application of the parallelogram law can be carried out in
order to obtain a resultant force.
Example: If three forces F1, F2 and F3 act at a point O, we
can find the resultant of any two forces and then add it to the
third force, using the parallelogram law.
FR = (F1 + F2) + F3
Chapter2 - Force Vectors
Trigonometric Rule





The resultant of two forces can be found analytically
from the parallelogram rule by applying the cosine and
the sine rules.
Redraw a half portion of the parallelogram to illustrate
the triangle head-to-tail addition of the components.
From this triangle, the magnitude of the resultant force
can be determined using the law of cosines and the
magnitude of the two forces components are determined
from the law of sines.
Cosine law: C  A 2  B 2  2 AB cos c
Sine law:
A
B
C


sin a sin b sin c
Chapter2 - Force Vectors
Example 1

The screw eye in the figure below is subjected to two
forces, F1 and F2. Determine the magnitude and
direction of the resultant force.
Chapter2 - Force Vectors
Answer 1

The magnitude and direction of the resultant force FR:

Cosine law:

Sine law:
F 
R
100  150
2
2
 2100150cos115  212.6 N
150
212.6
150

 sin  
sin 115  0.639
sin  sin 115
212.6
  39.8
  39.8  15  54.8
Chapter2 - Force Vectors
Example 2

Determine the magnitude of the component force F in
the figure below and the magnitude of the resultant force
FR if FR is directed along positive y axis.
Chapter2 - Force Vectors
Answer 2

FR is directed along positive y axis:

The magnitude of F and FR can be determined by
applying the law of sine:
F
200
sin 60

 F  200
 245 N
sin 60 sin 45
sin 45
F
200
sin 75

 F  200
 273N
sin 75 sin 45
sin 45
R
R
Chapter2 - Force Vectors
Example 3

It is required that the resultant force acting on the
eyebolt in the figure below be directed along the positive
x axis and that F2 have a minimum magnitude.
Determine this magnitude, the angle 
and the
corresponding resultant force.
Chapter2 - Force Vectors
Answer 3


FR is directed along the positive x axis.
The magnitude of F2 is minimum when its line of action
is perpendicular to the line of action of FR. This is when:
  90 
F2
F1
F
cos 60  R
F1
sin 60 
F2  800sin 60  693N
FR  800 cos 60   400 N
Chapter2 - Force Vectors
Addition of a System of Coplanar Forces

In many problems, it is desirable to resolve force F into
two perpendicular components in the x and y directions.
The components are called rectangular vector
components.

For analytical work there are two notations:
- Scalar Notation or - Cartesian Vector Notation
y
Fy
F

Fx
x
Chapter2 - Force Vectors
Scalar Notation:

We write the force F as (Fx, Fy) where Fx and Fy are the
scalar components of the force F in the directions of the
positive x and y axes, respectively.

If Fx and Fy are negative, it means that | Fx | and | Fy |
are directed along the negative x and y axes,
respectively.
Chapter2 - Force Vectors
Scalar Notation:

The magnitudes of the force two components are:
Fx  F cos 
Fy  F sin 
using a small slop triangle
Fx a

F c
Fy
b

F
c

a
Fx  F  
c

b
Fy   F  
c
Chapter2 - Force Vectors
Cartesian Vector Notation:

It is possible to represent the x and y components of a
force in terms of Cartesian unit vectors i and j, where i
and j represent the positive direction of the x and y axes,
respectively:
F = Fx i+ Fy j
where Fx and Fy are the scalar components of F
Chapter2 - Force Vectors
Coplanar Force Resultants

The resultants of several forces can be determined using
either the Cartesian vector notation or the scalar
notation. Example F1, F2 and F3:

Cartesian vector notation:
F1 = F1x i+ F1y j
F2 = – F2x i+ F2y j
F3 = F3x i – F3y j
The resultant is given by:

F = FRx i+ FRy j
Chapter2 - Force Vectors
Coplanar Force Resultants

Cartesian vector notation:
The vector resultant FR is then:
FR = F1 + F2 + F3
= (F1x i+ F1y j) + (– F2x i + F2y j) + (F3x i – F3y j)
= (F1x i – F2x i+ F3x i) + (F1y j + F2y j – F3y j)
= (F1x – F2x + F3x)i + (F1y + F2y – F3y )j
= (FRx)i + (FRy )j

Scalar notation:
FRx = F1x – F2x + F3x
FRy = F1y + F2y – F3y
These are the same as the i and j components of FR
Chapter2 - Force Vectors
Coplanar Force Resultants

In the general case, the x and y components of the
resultant of any number of coplanar forces can be
represented by:
FRx   Fx
FRy   Fy
FR  FRx  FRy
2
  tan
1
2
FRy
FRx
Chapter2 - Force Vectors
Example 4

Determine the x and y components of F1 and F2 acting
on the boom shown below. Express each force as a
Cartesian vector.
Chapter2 - Force Vectors
Answer 4

x and y components of F1:
F1x  200 sin 30   100 N  100 N 
F1 y  200 cos 30  173 N  173 N 

Cartesian Vector Notation of F1 : F1   100 i  173 j N
Chapter2 - Force Vectors
Answer 4

x and y components of F2:
F2 x 12
 12 

 F2 x  260   240 N  240 N 
260 13
 13 
F
5
5

 F  260   100 N  100 N 
260 13
 13 
2y
2y

Cartesian Vector Notation of F2 :
F2  240 i  100 j N
Chapter2 - Force Vectors
Example 5

The link shown below is subjected to two forces F1 and
F2. Determine the magnitude and direction of the
resultant force.
Chapter2 - Force Vectors
Solution 5

Scalar Notation: We resolve each force into its x and y
components, then we sum these components
algebraically:
Chapter2 - Force Vectors
Solution 5

Cartesian Vector Notation: Each force is first expressed
as a Cartesian Vector:
Chapter2 - Force Vectors
Cartesian Vectors

A Cartesian coordinate system is often used to solve
problems in 3 dimensions (3D).

The coordinate system is right-handed: The thumb of the
right hand points in the direction of the positive z axis
when the right hand fingers curled about this axis and
directed from the positive x towards the positive y axis.
Chapter2 - Force Vectors
Rectangular Components of a Vector

A vector A may have one, two or three rectangular
components along the x, y, z coordinates.

Two applications of the parallelogram law:
A = A’ + Az and A’ = Ax + Ay
A = Ax + Ay + Az
Cartesian Unit Vectors

In 3D, the set of Cartesian unit vectors,
i, j, k, is used to designate the directions
of the x, y, z axes, respectively.
Chapter2 - Force Vectors
Cartesian Vector Representation

Any vector A with scalar
components Ax, Ay and Az can be
written in the Cartesian vector form
as:
A = Ax i + Ay j + Az k
Magnitude of a Cartesian Vector

The magnitude of the vector A is
given by:
A  Ax  Ay  Az
2
2
2
Chapter2 - Force Vectors
Direction of a Cartesian Vector

The direction of vector A is defined by the angles α, β, and
γ measured between the tail of A and the positive x, y, z
axes located at the tail of A.

The angles α, β, and γ are found from their direction
cosines:
Ax
cos  
A
cos  
Ay
A
cos  
Az
A
cos 2   cos 2   cos 2   1

This means that only two of the angles
α, β, and γ have to be specified, the third
can be found from: cos 2   cos 2   cos 2   1
Chapter2 - Force Vectors
Direction of a Cartesian Vector
Ay
A Ax
Az
uA  
i
j
k uA is a unit vector in the direction of A
A A
A
A
u A  cos  i  cos  j  cos  k
A  Au A
A  A cos  i  A cos  j  A cos  k  Ax i  Ay j  Az k
Chapter2 - Force Vectors
Direction of a Cartesian Vector


Sometimes, the direction of A can be specified using two
angles,  and  .
By applying trigonometry yields to:
Az  A cos 
A  A sin 
Ax  A cos   A sin  cos 
Ay  A sin   A sin  sin 

Therefore A can be expressed in Cartesian vector form as:
A  A sin  cos  i  A sin  sin  j  A cos  k
Chapter2 - Force Vectors
Addition and Subtraction of Cartesian Vectors

The addition or subtraction of two or more vectors are
simplified if the vectors are expressed in terms of their
Cartesian components.

To find the resultant of a concurrent force system, express
each force as a Cartesian vector and add all the i, j, k
components of all forces in the system.

Example: add vector A to vector B
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
R=A+B
= (Ax +Bx) i + (Ay +By) j + (Az +Bz) k
In general:

FR   F   Fx i   Fy j   Fz k
Chapter2 - Force Vectors
Example 5

Express the force F shown in the figure below as a
Cartesian vector.
Chapter2 - Force Vectors
Answer 5

  60    45  ?
F = 200 N
cos 2   cos 2   cos 2   1
cos 2   cos 2 60   cos 2 45  1
cos   1  0.5  0.707  0.5
2
2
  cos 1 0.5  60 
F  F cos  i  F cos  j  F cos  k

 
 

F  200 cos 60 i  200 cos 60 j  200 cos 45 k
F  100 i  100 j  141.4 k N
Chapter2 - Force Vectors
Example 6

Determine the magnitude and the coordinate direction
angles of the resultant force acting on the ring shown
below.
Chapter2 - Force Vectors
Answer 6

The magnitude of FR:
FR   F  F1  F2
FR  60 j  80 k  50 i  100 j  100 k
FR  50 i  40 j  180 k N
FR 

502   402  1802
 191 N
The coordinate direction angles  ,  ,  are determined
from the components of the units vector acting in the
direction of FR :
u FR 
FR 50
40
180

i
j
k  0.2617 i  0.2094 j  0.9422 k
FR 191 191
191
cos   0.2617   74.8
cos   0.2094   102 
cos   0.9422   19.6 
Chapter2 - Force Vectors
Position Vectors

The position vector r is defined as a fixed vector which
locates a point in space relative to another point.

If r extends from the origin of coordinates O to a point
P(x, y, z) then:
r = xi + yj + zk
Chapter2 - Force Vectors
Position Vectors: General Case

More generally, the position vector may be directed from
point A to point B in space.

The vector position is denoted by r (or rAB).
by the head-to-tail vector addition we have: rA + r = rB

r = rB – rA = (xBi + yBj + zBk) – (xAi + yAj + zAk)
r = (xB – xA ) i + (yB – yA) j + (zB –zA) k
Chapter2 - Force Vectors
Force Vector Directed Along a Line

A force F (with magnitude F) acting in the direction of a
line represented by a position vector r (which is defined
by the unit vector u) can be expressed in Cartesian form
as:


r
 
F  Fu  F    F 
r


x B  x A  i   y B  y A  j  z B  z A  k
 x B  x A 2   y B  y A 2   z B  z A  2
Chapter2 - Force Vectors





Dot Product

The dot product of vectors A and B is defined as the
product of the magnitudes of A and B and the cosine of
the angle  between their tails:
A  B  AB cos 

The dot product is used to determine:
◦ The angle between two vectors.
◦ The projection of a vector in a specified direction.

The dot product between two vectors yields a scalar
Chapter2 - Force Vectors
Dot Product: Law of Operation
AB  B A

Commutative law:

Multiplication by a scalar: aA  B  aA  B  A  aB

Distributive law:
A  B  D  A  B  A  D
Cartesian Vector Formulation:
A B  Ax Bx  Ay By  Az Bz
The Angle Formed Between Two Vectors:
 AB

AB


  cos 1 
Chapter2 - Force Vectors