SCALARS AND VECTORS
All physical quantities in engineering mechanics are
measured using either scalars or vectors.
Scalar. A scalar is any positive or negative physical
quantity that can be completely specified by its
magnitude. Examples of scalar quantities include
length, mass, time, density, volume, temperature,
energy, area, speed.
VECTOR
A vector is any physical quantity that requires both a magnitude and
a direction for its complete description. Examples of vectors
encountered in statics are force, position, and moment. A vector is
shown graphically by an arrow. The length of the arrow represents the
magnitude of the vector, and the angle q between the vector and a
fixed axis defines the direction of its line of action. The head or tip of
the arrow indicates the sense of direction of the vector
TYPES OF VECTORS
Physical quantities that are vectors fall into one of the three classifications as
free, sliding or fixed.
A free vector is one whose action is not confined to or associated with a
unique line in space. For example if a body is in translational motion, velocity
of any point in the body may be taken as a vector and this vector will describe
equally well the velocity of every point in the body. Hence, we may represent
the velocity of such a body by a free vector.
In statics, couple moment is a free vector.
A sliding vector is one for which a unique line in space must be
maintained along which the quantity acts. When we deal with the external
action of a force on a rigid body, the force may be applied at any point
along its line of action without changing its effect on the body as a whole
and hence, considered as a sliding vector.
A fixed vector is one for which a unique point of application is
specified and therefore the vector occupies a particular position in
space. The action of a force on a deformable body must be specified
by a fixed vector.
Principle of Transmissibility (Taşınabilirlik İlkesi)
The external effect of a force on a rigid body will remain
unchanged if the force is moved to act on its line of action.
Equality and Equivalence of Vectors
Two vectors are equal if they have the same dimensions, magnitudes and directions.
Two vectors are equivalent in a certain capacity if each produces the very same effect
in this capacity.
Addition of Vectors is done according to the parallelogram principle of
vector addition. To illustrate, the two “ component ” vectors 𝐴 and 𝐵 are
added to form a “ resultant ” vector 𝑅.
R
A
B
RBA
Parallelogram law
RBA
R
A
B
Triangle law
Subtraction of Vectors is done according to the parallelogram law.
Multiplication of a Scalar and a Vector
VaUaVUa UbUaUba
UabUba UaUa
BABAR
R
A B
B
Vector Addition of Forces
Experimental evidence has shown that a force is a vectorquantity since it has a specified magnitude, direction, and senseand it adds according to the parallelogram law. Two commonproblems in statics involve either finding the resultant force,knowing its components, or resolving a known force into twocomponents.
Finding the Components of a Force.
Finding a Resultant Force.
If more than two forces are to be added, successiveapplications of the parallelogram law can be carriedout in order to obtain the resultant force.
Addition of Several Forces
Vector Components and Resultant Vector Let the sum of 𝐴 and 𝐵 be 𝑅. Here, 𝐴 and 𝐵
are named as the components and 𝑅 is named as the resultant.
sinsinsin
RBA
cos2222 ABBAR
Sine theorem
Cosine theorem
RBA
R
A
B
R
A
B
q
q
qcos2222 ABBAR
Cosine theorem
Note that
(Magnitude of the resultant force can be determined using the law of cosines, and itsdirection is determined from the law of sines.)
Example-1: The screw eye is subjected to two forces, 𝐹1 and 𝐹2.Determine the magnitude and direction of the resultant force.
Example-2: Resolve the horizontal 600-N force in thefigure into components acting along the u and v axesand determine the magnitudes of these components.
600-N
Example-3: It is required that the resultant force acting on the
eyebolt be directed along the positive x axis and that 𝐹2 have aminimum magnitude. Determine this magnitude, the angle q, andthe corresponding resultant force.
The relationship between a force and its vector components must
not be confused with the relationship between a force and its
perpendicular (orthogonal) projections onto the same axes.
For example, the perpendicular projections of force onto axes a
and b are and , which are parallel to the vector components of
and .
F
aF
bF
1F
2F
F
a
b
//a
//b
1F
2F
F
a
b
a
b
aF
bF
Components: F1 and F2 Projections: Fa and Fb
It is seen that the components of a vector are not necessarily equal to
the projections of the vector onto the same axes. The components and
projections of are equal only when the axes a and b are
perpendicular.
F
F
a
b
//a
//b
1F
2F
F
a
b
a
b
aF
bF
Components: F1 and F2 Projections: Fa and Fb
Example-4: The vector 𝑉 lies in the plane defined by the intersecting lines LA and LB. Its
magnitude is 400 units. Suppose that you want to resolve 𝑉 into vector components
parallel to LA and LB. Determine the magnitudes of the vector components.
80°
60°
V
LA
LB
Example-5: Determine the projections Pa and Pb of 𝑉 onto the lines LA and LB.
Unit Vector A unit vector is a free vector having a magnitude of 1 (one) as
eornU
U
U
Un
It describes direction. The most convenient way to describe a vector in a certain
direction is to multiply its magnitude with its unit vector.
nUU
U
1
U
n
and U have the same unit, hence the unit vector is dimensionless. U
CARTESIAN COORDINATES Cartesian Coordinate System is
composed of 90° (orthogonal) axes. It consists of x and y axes in two
dimensional (planar) case, x, y and z axes in three dimensional (spatial)
case. x-y axes are generally taken within the plane of the paper, their
positive directions can be selected arbitrarily; the positive direction of
the z axes must be determined in accordance with the right hand rule.
Cartesian Unit Vectors In three dimensions, the set of
Cartesian unit vectors, 𝑖 , 𝑗, 𝑘, is used to designate the
directions of the x, y, z axes, respectively.
Vector Components in Two Dimensional (Planar) Cartesian Coordinates
jUiUU
jUU
iUU
yx
yy
xx
x
y
yx
U
U
UUU
qtan
22
x
y
U
i
j
yU
xU
q
yx UUU
Vector Components in Three Dimensional (Spatial) Cartesian Coordinates
unit vector along the x axis, ,
unit vector along the y axis, ,
unit vector along the y axis, ,
ji
k
222
zyx
zyx
UUUU
kUjUiUU
x
y
z
U
i
j
k
zU
yU
xU
In three dimensional case
kzzjyyixxr ABABABB/A
A (xA, yA, zA)
B (xB, yB, zB)
x
y
z
B/Ar
i
jk
Position Vector: It is the vector that describes the location of one point with respect
to another point.
In two dimensional case
jyyixxr ABABB/A
y
i
j
A (xA, yA)
B (xB, yB)
B/Ar
x
Example-6: An elastic rubber band is attached to points Aand B. Determine its length and its direction measuredfrom A toward B.
* When the direction angles of a force vector are given;
The angles, the line of action of a force makes with the x, y and z axes are named as
direction angles. The cosines of these angles are called direction cosines; they
specify the line of action of a vector with respect to coordinate axes.
In this case, direction angles are qx, qy and qz.
Direction cosines are cos qx, cos qy and cos qz.
cos qx = l cos qy = m cos qz = n
222
cos
cos
cos
zyx
zz
yy
xx
FFFF
FF
FF
FF
q
q
q
kjiFF
kFjFiFF
kFjFiFF
zyx
zyx
zyx
qqq
qqq
coscoscos
coscoscos
11coscoscos
1
coscoscos
222222
2
222
2
22222
nml
F
FFF
F
FFFFF
knjmiln
kjin
zyx
zyx
zyx
F
zyxF
qqq
qqq
kjiFF zyx
qqq coscoscos
Example-7: Express the force as a Cartesian vector.
2
12
2
12
2
12
121212
zzyyxx
kzzjyyixxF
AB
ABFF
nFF F
* When coordinates of two points along the line of action of a force are given;
Example-8: The force acts on the hook. Express itas a Cartesian vector.
* When two angles describing the line of action of a force is given;
sincosFsinFF
coscosFcosFF
sinFF
cosFF
xyy
xyx
z
xy
First resolve F into horizontal and vertical components.
Then resolve the horizontal component Fxy into x- and y-components.
Example-9: Express the forces shown in the figures as a Cartesian vector.
F=50 N
Addition of Cartesian Vectors
jVUiVUjViVjUiUVU
jViVVjUiUU
yyxxyxyx
yxyx
kVjViVV zyx
In two dimensional case
In three dimensional case
kUjUiUU zyx
kVUjVUiVUVU zzyyxx
Example-10: Two forces act on the hook shown.
Specify the magnitude of 𝐹2 and its coordinate
direction angles so that the resultant force 𝐹R
acts along the positive y axis and has amagnitude of 800 N.
Dot (Scalar) Product A scalar quantity is obtained from the dot product of two vectors.
VU
VUcos cosVUVU
aUV irrelevant is tionmultiplica of order
aVU
zzyyxx
zyxzyx
VUVUVUVU
kVjViVV kUjUiUU
ik ,kj ,cosjiji
kk ,jj ,cosiiii
00090
1110
U
V
In terms of unit vectors in Cartesian Coordinates;
//
//
UUU
nnUU
Normal and Parallel Components of a Vector with respect to a Line
cos // qUU Magnitude of parallel component:
Normal (Orthogonal) component:
n
U
U
//U
q
Parallel component:
nUUUnUnU
//
1
, coscos qq
Example-11: The frame shown is subjected to a horizontal force 𝐹 = 300 𝑗. Determinethe magnitude of the components of this force parallel and perpendicular to member AB.
𝐹 = 300 𝑗
Cross (Vector) Product: The multiplication of two vectors in cross product
results in a vector. This multiplication vector is normal to the plane containing the
other two vectors. Its direction is determined by the right hand rule. Its magnitude
equals the area of the parallelogram that the vectors span. The order of
multiplication is important.
YUVUYVU
VaUVUaVUa
VU
VUsin sinVUVU
WUV , WVU
qqU
q
V
U
V
W
Wq
jk i ,ijk , kij
jik ,ikj , kji
sinjiji
kk ,jj ,siniiii
190
0000
In terms of unit vectors in Cartesian Coordinates;
kVUVUjVUVUiVUVUVU
VUkVUiVUj VUkVUjVUi
V
U
j
V
U
i
VVV
UUU
kji
VU
kVjViVkUjUiUVU
xyyxzxxzyzzy
xyyzzxyxxzzy
y
y
x
x
zyx
zyx
zyxzyx
x
y
z
i
j
k
i
j
k
i
j
k
+ +
Consider the straight lines OA and OB.
a) Determine the components of a unit vector that
is perpendicular to both OA and OB.
b) What is the minimum distance from point A to
the line OB?
A (10, -2, 3) m
y
xq
z
B (6, 6, -3) m
O
Example-12:
a) Determine the components of a unit vector that is perpendicular to both OA and OB.
A (10, 2, 3) m
q
B (6, 6, 3) m
O
b) What is the minimum distance from point A to the line OB?
OBr /
OAr /
kjirkjir OAOB
3210366 //
md
drrrr
rd
OBOA
r
OBOA
OA
71.9
36636.87sin
sin
9
222
////
/
q
q
Vector normal to both OA and OB is
kjir
ijikjkr
kjikjirrr OBOA
724812
18186123060
3663210//
Unit vector normal to both OA and OB is
kjinkji
r
rn
824.0549.0137.0724812
724812
36.87
222
Consider the triangle ABC.
a) What is the surface area of ABC?
b) Determine the unit vector of the outer normal of surface ABC.
c) What is the angle between AC and AB?
A (12, 0, 0) m
y
x
z
B (0, 16, 0) m
C (0, 0, 5) m
Example-13:
a)What is the surface area of ABC?
A (12, 0, 0) m
y
x
z
B (0, 16, 0) m
C (0, 0, 5) mB
A
C
c) What is the angle between AC and AB?
kiACjiAB
5121612
AC
AB
ikjACAB
kijiACAB
8019260
5121612
2222
24.1082
1926080mABCs
b) Determine the unit vector of the outer normal of surface ABC.
kjikji
ACAB
ACABn
887.0277.0369.0
48.216
1926080
368.56cos1320144
cos51216125121612
cos
2222
kiji
ACABACAB