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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 are 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
P
q
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 has a unique line of action in space but not a unique point
of application. 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 thus, it is a sliding vector.
A fixed vector is one for which a unique point of application is
specified. The action of a force on a deformable body must be
specified by a fixed vector. In this instance the forces and
deformations within the body depend on the point of application of
the force, as well as on its magnitude and line of action.
P
q
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.
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, andits direction is determined from the law of sines.)
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
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.
x
y
z
z
z
x yx
y
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
* 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. (In this case, direction angles are qx, qy and qz)
The cosines of these angles are called direction cosines. (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
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
kFjFiFF
kFjFiFF
zyx
zyx
qqq coscoscos
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;
* When two angles describing the line of action of a force are 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.
Dot (Scalar) Product A scalar quantity is obtained from the dot product of two vectors.
VU
VU
VUVU
aUV
irrelevant is tionmultiplica of order
aVU
cos
cos
zzyyxx
zyxzyx
VUVUVUVU
kVjViVV kUjUiUU
ik ,kj ,cosjiji
kk ,jj ,cosiiii
00090
1110
U
V
In terms of unit vectors in Cartesian Coordinates;
n nUU
//
Normal and Parallel Components of a Vector with respect to a Line:
UU qcos//
Magnitude of parallel component:
Normal (Orthogonal) component:
n
U
U
//U
q
Parallel component of vector 𝑼 to a line may be determined using dot product:
nUU
UnUnU
//
1
coscos qq
//UU U
𝑫𝒐𝒕 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒗𝒆𝒄𝒕𝒐𝒓 𝑼 𝒂𝒏𝒅 𝒖𝒏𝒊𝒕 𝒗𝒆𝒄𝒕𝒐𝒓 𝒏 is
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.
VU
VU
VUVU
WUV ,WVU
q
q
sin
sinU
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;
x
y
z
i
j
k
i
j
k
i
j
k
+ +
