Download Chapter-5 Vector

Comprehensive Physics Notes
Class X
Chapter-5 Vector
ABDUL RASHEED
Educast
Learnig Objectives
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Introduction
Physical quantities
Vector quantities
Scalar quantities
Vector representation
Multilpication of a vector by a number
Negative of a vector
Trigonomatric Ratio
Resolution of vectors
SCALARS & VECTORS
Physical Quantities:
All those quantities which can be measured are called physical quantities. Physical Quantities can
be measured by means of magnitude and units.
Magnitude:
A symbol which gives us the quantity of substance is called magnitude.
Unit:
A symbol which gives its relation is called unit. e.g.
5 KG 5: Magnitude Kg: Unit
10 N 10: Magnitude N : Unit
Type of Physical Quantities:
Physical quantities are of two types.
(i) Scalar quantities (ii) Vector Quantities
SCALAR QUANTITIES
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 deirection.
vector quantities are expressed by using bold letters with arrow sign such as:
vector quantities can not 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 choosen in the given direction.
Negative of a vector:
Negative vector is defined as:
“A vector just equal in magnitude but exactly opposite in direction is called negative of a
vector.”
OR
“A vector having the same magnitude as that of a given vector but opposite in direction is called
negative of a vector.”
ADDITION OF VECTORS
Addition of Vectors by Head to Tail Rule:
Consider two vectors A and B represented by lines OP and OQ respectively as shown in figure we
can and these two vectors by placing the tail of second vector to the head of first vector.
Now join the tail of first vector to the head of last vector, it gives us the Resultant vector as shown
in Fig.
If more than two vectors are to be added than same method will be adopted for that. First draw the
first vector than place the second vector so that its tail is at the head of First vector. Now add all the
vectors just by joining the head of first vector to the tail of Next vector and in last join the tail of
first vector to the head of last vector it gives us Resultant vector as shown in figure.
Resultant Vector:
“A vector which joins the tail of first vector to the head of last vector is called resultant Vector.”
OR
“A single vector which gives the combined effect of all the vectors which are to be added is called
resultant vectors.”
It can be represented by the following equation:
R = A + B + C + ………
Where R is the resultant vector, while A, B, C, are vectors to be added.
Subtraction of Vector:
Vectors cannot be subtracted directly. They are subtracted by means of addition. To subtract vector
from another vector sign of the vector is changed and then added to the other vector. For example
if a vector B is to be subtracted from a Vector: A then A-B is found by adding vector A and –B.
Subtraction of vectors can be illustrated as follows:
Trigonometry:
Trigonometry is an important branch of mathematics and is used to solve various problems in
physics. Considering the right angle triangle ABC, angle <ABC is right angle and <BAC is denoted
Base: (BC) Side adjacent to the angle Ø is called Base.
Perpendicular: (AB) Side opposite to the angle 0 is called Perpendicular.
Hypotenuse: (AC) Largest side or the side opposite to the right angle is called Hypotenuse
The ratios between any two sides of the right triangle represented by different names. Some of
important ratios are as follows:
Sin Ø = Prep / Hyp Cosec Ø = Hyp / Perp
Cos Ø = Base / Hyp Sec Ø = Hyp / Base
Tan Ø = Perp / Base Cot Ø = Base / Perp
The values of trigonometric ratios are changed if Ø is changed.
Resolution of Vector:
The process of splitting a vector into its parts (components) is called resolution of a vector.
Generally a vector is resolved into two components at right angle to each other Such components
are called rectangular components.
Horizontal Component: The component which is along horizontal direction is called horizontal
component.
Vertical Component: The Component which is along vertical direction is called vertical Component
Consider a vector F, which shows the representative line AB making angle Ø with x-axis From B
draw perpendicular BC on x-axis. Suppose AB and AC are represented by two Vectors. Vector BC is
parallel to y-axis and y Vector AC is along x-axis. Hence we denote Vector AC by Fx and vector AB by
Fy by Applying head to tail rule of vector addition, The sum of vectors Fx and Fy is equal to F.
Therefore, FX and FY are rectangular components B of Vector F.
The magnitude of these components can be Fy determined by using trigonometric ratios.
Now considering right angle triangle ABC. Horizontal
Vertical component / YComponent / X-Component
Component
Cos Ø = Base/Hyp
Sin Ø = Perp/Hyp
Cos Ø = AC/BC
Sin Ø = AB / BC
Cos Ø = Fx / F
Sin Ø = Fy / F
Fx = FCos Ø
Fy = FSin Ø
Addition of Rectangular Components of Vectors: OR
Composition of Vector:
Rectangular components of vector (components that are perpendicular to each other) can be
joining together to form resultant vector or original vector.
Considering right angle triangle ABC. Where: Fx = AB = Base , Fy = BC = Perp, F = AC = Hyp
For magnitude of vector using Pythagoras theorem.
(H)2 = (B)2 + (P)2
(AC)2 = (AB)2 + (BC)2
AC = √(AB)2 + (BC)2
F = √Fx2 + Fy2
For direction of vector using trigonometric ratio:
Tan Ø = Perp / Base
Tan Ø = BC / AC
Tan Ø = Fy / Fx
0 = Tan-1 (Fy / Fx)
PARALLELOGRAM LAW
OF VECTOR ADDITION
Acccording to the parallelogram law of vector addition:
"If two vector quantities are represented by two adjacent sides or a
parallelogram
then the diagonal of parallelogram will be equal to the resultant of these two
vectors."
EXPLANATION
Consider two vectors
. Let the vectors have the following orientation
parallelogram of these vectors is :
According to parallelogram law:
MAGNITUDE OF
RESULTANT VECTOR
Magintude or resultant vector can be determined by using either sine law or cosine law.
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
components of vector
.
Vx = Horizontal component of
Vy = Vertical component of
.
.
MAGNITUDE OF
HORIZONTAL
COMPONENT
Consider right angled triangle 
.Thus Vx and Vy are the rectangular
MAGNITUDE OF
VERTICAL COMPONENT
Consider right angled triangle 
Addition of vectors by Head to Tail method (Graphical Method)
Head to Tail method or graphical method is one of the easiest method 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
coincides with the
STEP # 3
Join 'O' and 'B'.
represents resultant vector of given vectors
and
i.e.
STEP # 4
Measure the length of line segment
get the magnitude of resultant vector.
STEP # 5
and multiply it with the scale choosen initially to
The direction of the resultant vector is directed from the tail of vector
vector
.
to the head of