Definition:- There
are two types of physical quantities scalar quantities and vector quantities, these
are given below:
Scalar: - A
scalar quantity is one that is defined by a single number with appropriate
units.
Ex: length, area, volume, mass and time.
Vector: - A
vector quantity is defined completely when we know both its magnitude (with units) and its direction of application. For example
force, velocity and acceleration.
Examples: Two
very simple example demonstrating the differences between scalars and vectors
are:
A speed
of 10 km/h is a scalar quantity
A
velocity of 10 km/h is a vector quantity.
**
A
vector must also satisfy some other basic rules of combination (addition, multiplication
etc.). For example angular
displacement is a quantity that has
both direction and magnitude but
it does NOT obey the addition rule – so
it is NOT a vector.
Vector notation: - A vector can be written down in many ways. Common notation of a vector
 |
| Vector Notation |
Vector magnitude is nothing but length
of line OA.
Types of Vectors:-
Unit Vector: -A vector whose modulus is 1 is called a unit vector, denoted as ȃ and it is defined as:
 |
| Unit Vectors |
The unit vectors of i, j and k:-
i
= (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)
A vector can be
expressed in terms of its component x, y, z w. r. t. the unit vector (xi, yj,
zk).
Ex: vector a = (3, -1, 4), find modulus and unit
vector of a.
Equal vectors: -Two
vectors a and b are equal if they have the same magnitude and direction
if vector a = vector b
Means: a1 = b1 , a2 = b2 and a3 = b3
Parallel vector: - Two vectors are called parallel vector if a = λb where If λ is a scalar quantity,
λ > 0, the
vector a is in the same direction as
b and has magnitude λb.
λ < 0, the
vector a is in the opposite
direction as b and has magnitude λb.
In component form
a1 =
λb1 , a2 = λb2 and a3 = λb3
Means vectors are:
Parallel if λ > 0, and Anti-parallel if λ < 0.
Ex: a boat steams
at 4 towards east for one hour. The tide is running north-east where will the
boat be after one hour?
a = velocity of boat b = velocity of tide, The net velocity is represented by OC.
 |
| Parallel Vectors |
Addition laws: -
Parallelogram rule: - the
sum or resultant of two vectors a
and b is found by forming a
parallelogram with a and b as two adjacent sides:
 |
| parallelogram law |
if a = (a1, a2, a3) and b = (b1, b2, b3) then sum of vector a and b is defined as
a + b = (a1 + b1, a2
+ b2, a3 + b3)
Triangle law: if two vectors a and b are represented in magnitude and direction by two sides of a triangle taken in order, then their sum is represented in magnitude and direction by the closing third side.
 |
| triangle law |
** If two or more than two
vectors are joining sequentially then their resultant vector is represented by
the line joining initial and final position and value of resultant is the sum
of all joining vectors.
 |
| Sum of Vectors |
Above figure shows polygon law: so resultant r vector is equal to
r = a + b + d + e
Properties of vector addition: -
i.
a
+ b = b + a
(commutative law) Ã order is NOT important
ii.
(a
+ b) + c = a + (b + c)
(Associative law)
iii. λ(a + b) = λa + λb (distributive law)
Subtraction of vector: -
Subtraction of two vectors is calculated as,
a – b = (a1 – b1, a2
– b2, a3 – b3,)
Ex: a = (2, 1, 0), b = (-1, 2, 3) and c = (1, 2,
1) then find
·
a + b
·
2 a – b
·
a + b – c =
·
unit vector of c
0 Comments