Post on 10-Feb-2020
Scalars and Vectors
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 forced 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.
To sum up, the equality of two vectors is
determined by the vectors themselves, but the
equivalence between two vectors is determined
by the situation at hand.
PROPERTIES OF VECTORS
Addition of Vectors is done according to the parallelogram
law of vector addition.
Subtraction of Vectors is done according to the parallelogram law.
Multiplication of a Scalar and a Vector
Unit Vector A unit vector is a free vector having a magnitude of 1 (one) as
U
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
Cartesian Coordinates Cartesian coordinate system is composed of 90°
(orthogonal) axes. It consists of x and y axes in two dimensional (planer) 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 (Planer) Cartesian Coordinates
,
jVUiVUjViVjUiUVUjViVV
jUiUUjUUiUU
yyxxyxyxyx
yxyyxx
Vector Components in Three Dimensional (Spatial) Cartesian Coordinates
kVUjVUiVUVU
kVjViVV
UUU UkUjUiUU
zzyyxx
zyx
zyxzyx
222
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
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 equal to 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
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