SCALARS & VECTORS
SCALAR QUANTITIES www.amuqeet.angelfire.com
Physical quantities which can completely be specified by a number (magnitude)
having an appropriate unit are known as "SCALAR QUANTITIES".
Scalar quantities do not need direction for their description.
Scalar quantities are comparable only when they have the same physical dimensions.
Two or more than two scalar quantities measured in the same system of units are equal
if they have the same magnitude and sign.
Scalar quantities are denoted by letters in ordinary type.
Scalar quantities are added, subtracted, multiplied or divided by the simple rules of
algebra.
EXAMPLES
Work, energy, electric flux, volume, refractive index, time, speed, electric potential,
potential difference, viscosity, density, power, mass, distance, temperature, electric
charge, electric flux etc.
VECTORS
QUANTITIES
Physical quantities having both magnitude and direction
with appropriate unit are known as "VECTOR QUANTITIES".
We can't specify a vector quantity without mention of direction.
vector quantities are expressed by using bold letters with arrow sign such as:
vector quantities cannot be added, subtracted, multiplied or divided by the simple rules
of algebra.
vector quantities added, subtracted, multiplied or divided by the rules of trigonometry
and geometry.
EXAMPLES
Velocity, electric field intensity, acceleration, force, momentum, torque, displacement,
electric current, weight, angular momentum etc.
REPRESENTATION OF
VECTORS
On paper vector quantities are represented by a straight line with arrow head pointing
the direction of vector or terminal point of vector.
A vector quantity is first transformed into a suitable scale and then a line is drawn with
the help of the
scale chosen in the given direction.
RESOLUTION OF VECTOR
DEFINITION
The process of splitting a vector into various parts or components is called
"RESOLUTION OF VECTOR"
These parts of a vector may act in different directions and are called "components of
vector".
We can resolve a vector into a number of components .Generally there are three
components of vector viz.
Component along X-axis called x-component
Component along Y-axis called Y-component
Component along Z-axis called Z-component
Here we will discuss only two components x-component & Y-component which are
perpendicular to each other. These components are called rectangular components of
vector.
METHOD OF
RESOLVING
A VECTOR INTO
RECTANGULAR
COMPONENTS
Consider a vector
acting at a point making an angle with positive X-axis. Vector
is
represented by a line OA.From point A draw a perpendicular AB on X-axis. Suppose
OB and BA
represents two vectors. Vector OA is parallel to X-axis and vector BA is parallel to Yaxis. Magnitude
of these vectors are Vx and Vy respectively. By the method of head to tail we notice
that the sum of these vectors is equal to vector
rectangular components of vector
.
Vx = Horizontal component of
.
Vy = Vertical component of
.
.Thus Vx and Vy are the
MAGNITUDE OF
HORIZONTAL
COMPONENT
Consider right angled triangle 
MAGNITUDE OF
VERTICAL COMPONENT
Consider right angled triangle 
Magnitude to vector V can be find according to Pythagoras theorem
OB2 + AB2= OA2
VX2 + VY2 = V2
V =√ VX2 + VY2
MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)
MULTIPLICATION
OF A VECTOR
BY A SCALAR
When a vector is multiplied by a positive number (for example 2, 3 ,5, 60 unit etc.)
or a scalar only its magnitude is changed but its direction remains the same as that
of the original vector.
If however a vector is multiplied by a negative number (for example -2, -3 ,-5, -60
unit etc.) or a scalar not only its magnitude is changed but its direction also
reversed.
Addition of vectors by Head to Tail method (Graphical Method)
Head to Tail method or graphical method is one of the easiest methods used to find
the resultant vector of two of more than two vectors.
DETAILS OF
METHOD
Consider two vectors
and
acting in the directions as shown below:
In order to get their resultant vector by head to tail method we must follow the
following steps:
STEP # 1
Choose a suitable scale for the vectors so that they can be plotted on the paper.
STEP # 2
Draw representative line
of vector
Draw representative line
of vector
head of vector
such that the tail of
.
STEP # 3
Join 'O' and 'B'.
represents resultant vector of given vectors
and
i.e.
coincides with the
STEP # 4
Measure the length of line segment
and multiply it with the scale chosen initially to
get the magnitude of resultant vector.
STEP # 5
The direction of the resultant vector is directed from the tail of vector
of vector
to the head
.
Negative vector: A vector equal in magnitude but opposite in direction to a
vector is called negative vector.
Exercise QUESTIONS
5.2)Ans: Magnitude of resultant vector A AND B A+ B WHEN THEY ARE IN SAME
DIRECTION, AND if resultant of vector A and B IS ZERO THAT A+B =0 THAN B=-A
IT MEANS THAT VECTOR B MUST BE NEGATIVE VECTOR OF A
5.3 Ans: No to cancel two forces must be equal and opposite to each other.
5.4 Ans: Resultant of two vector will be zero if they are equal in magnitude and
opposite in direction.
5.5 Ans: Sum of three vectors will be zero if they form a closed figure or sum of
three vectors will be zero if sum of two vector s is equal and opposite to third
vector.
5.6 Ans: No if one component of a vector is zero then the vector will not be zero ,
such will either be along x-axis or along y-axis .
5.7 ans: The vectors A and B will be perpendicular to each other.
5.8 Ans: The body will move in the direction of C
5.9 Ans: When the body moves along a straight line then magnitude of
displacement is equal to the distance between two points.
5.11Ans: Total distance is sum of the distance to bazaar and back home i.e. 5+5 =
10 km
Total displacement will be zero because of same magnitude (distance) and opposite
direction.
5.12 fill in blanks 1.Scalar 2. Direction 3. Resolution 4. Zero 5. 17N , 3N