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SCALAR
A physical quantity that is completely characterized
by a real number (or by its numerical value) is called
a scalar. In other words, a scalar possesses only a
magnitude. Mass, density, volume, temperature, time,
energy, area, speed and length are examples to scalar
quantities.
VECTOR
Several quantities that occur in mechanics require a description in
terms of their direction as well as the numerical value of their
magnitude. Such quantities behave as vectors. Therefore, vectors
possess both magnitude and direction; and they obey the
parallelogram law of addition. Force, moment, displacement, velocity,
acceleration, impulse and momentum are vector quantities.
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
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. In
other words, a force may be applied at any point on its given line
of action without altering the resultant external effects on the
rigid body on which it acts.
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.
PROPERTIES OF VECTORS
Addition of Vectors is done according to the parallelogram law
of vector addition.
UVVU
MVUMVU or WVU
W
U
V
Subtraction of Vectors is done according to the parallelogram law.
Multiplication of a Scalar and a Vector
VaUaVUa UbUaUba
UabUba UaUa
ZVUVU
Z
U V
V
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. Therefore,
may be expressed in terms of both its magnitude and direction separately. U (a
scalar) expresses the magnitude and (a dimensionless vector) expresses the
directional sense of .
U
n
U
Vector Components and Resultant Vector Let the sum of
and be . Here, and are named as the components
and is named as the resultant.
U
V
W
U
V
W
sinsinsin
WVU
cos2222 UVVUW
Sine theorem
Cosine theorem
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 y x
y
Vector Components in Two Dimensional (Planar) Cartesian Coordinates
unit vector along the x axis, , unit vector along the y axis, j
i
jVUiVUjViVjUiUVU jViVV
jUiUU jUU iUU
yyxxyxyxyx
yxyyxx
x
y
yx
yx
U
Utan
UUU
UUU
22
x
y
U
i
j
yU
xU
Vector Components in Three Dimensional (Spatial) Cartesian Coordinates
kVUjVUiVUVU
kVjViVV
zzyyxx
zyx
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
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
In three dimensional case
kzzjyyixxr ABABABB/A
A (xA, yA, zA)
B (xB, yB, zB)
x
y
z
B/Ar
ij
k
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
nUUUnUnU
UU
//
1
//
, coscos
cos
Magnitude of parallel component:
Normal (Orthogonal) component:
n
U
U
//U
Parallel component:
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
U
V
U
V
W
W
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
+ +
Mixed Triple Product: It is used when taking the moment of a force about a line.
zyx
zyx
zyx
zyx
zyxzyx
zyx
zyx
zyx
WWW
VVV
UUU
WVU
or
WWW
VVV
kji
kUjUiUWVU
kWjWiWW
kVjViVV
kUjUiUU