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.
Vector Components in Two Dimensional (Planar) Cartesian Coordinates
unit vector along the x axis, , unit vector along the y axis, i
j
i
jVUiVUjViVjUiUVU jViVV
jUiUU jUU iUU
yyxxyxyxyx
yxyyxx
x
y
yxyx
U
U
UUUUUU
an t
22
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
Position Vector It is the vector that describes the location of one
point with respect to another point.
jyyixxr
yyrxxr
rrr
jrirrrr
ABAB
AByABx
yx
yxyx
B/A
B/AB/A
B/AB/AB/A
B/AB/AB/AB/AB/A
,
22
In two dimensional case
In three dimensional case
kzzjyyixxr
zzryyrxxr
rrrr
krjrirrrrr
ABABAB
ABzAByABx
zyx
zyxzyx
B/A
B/AB/AB/A
B/AB/AB/AB/A
B/AB/AB/AB/AB/AB/AB/A
, ,
222
Dot (Scalar) Product A scalar quantity is obtained from the dot product of two
vectors.
VU
VUVUVU
aUVaVU
cos cos
irrelevant istion multiplica oforder
zzyyxx
zyxzyx
VUVUVUVU
kVjViVVkUjUiUU
ikkjjiji
kkjjiiii
, ,
, ,
s,Coordinate Cartesian in vectors unit of terms In
00090cos
1110cos
U
V
Normal and Parallel Components of a Vector with respect to a Line
nUUUnUnU
UU
//
1
//
, coscos
cos
Magnitude of parallel component
Parallel component
Normal (Orthogonal) component
//
//
UUU
nnUU
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
VUVUVU
WUVWVU
sinsin
,
jkiijkkij
jikikjkjijiji
kkjjiiii
, ,
, , , 190sin
0 , 0 , 00sin
s,Coordinate Cartesian in orsunit vect of In terms
-- -
kVUVUjVUVUiVUVUVU
VUkVUiVUjVUkVUjVUi
V
U
j
V
U
i
VVV
UUU
kji
VU
kVjViVkUjUiUVU
xyyxzxxzyzzy
xyyzzxyxxzzy
y
y
x
x
zyx
zyx
zyxzyx