Scalars:
Those physical quantities which possess only magnitude and no direction are called scalars.
- It is specified completely by a single number, along with the proper unit.
- Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers i.e by using the rules of ordinary algebra.
- Some examples of vectors are mass, time, distance, speed, work, energy, power etc.
Vectors:
Those physical quantities which possess both magnitude and direction are called vectors.
- A vector is specified by giving its magnitude (a number along with its unit) and its direction.
- Some examples of vectors are displacement, velocity, acceleration momentum, force etc.
- Vectors can’t be added, subtracted, multiplied and divided just as the ordinary numbers i.e by using the rules of ordinary algebra.
- Vectors are added by using by using ‘Triangle Law of addition’ or by ‘parallelogram law of addition’.
- Some of the quantities, like current, are having both magnitude and direction but are not considered vector quantities, because they do not follow laws of addition.
‘A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition’.
Vectors and Scalars (Video)
( In Hindi + English Mix)
Types of Vector:
- Equal vectors: Two vectors having same magnitude as well as direction are known as equal vectors.
- Negative or Opposite vectors: A vector having same magnitude as that of a given vector and the direction opposite to the given vector is called the negative or opposite vector to the given vector.
- Collinear vectors: All those vectors which either act along the same line or act along parallel lines are called collinear vector.
- Co initial vectors: All those vectors which start from the same point are known as co-initial vectors.
- Co-terminus vector: All those vectors which terminate at the same point are known as co-terminus vector.
- Co planar vector: All those vectors which act along the same plane are called co planar vector.
Unit Vectors:
Unit vector along a given vector is a vector of unit magnitude and has the same direction as that of the given vector.
It is mathematically defined as
……….. (i)
………… (ii)
- Physically, unit vector represents the direction of the given vector.
- The unit vectors which are mutually perpendicular to each other are known as orthogonal unit vectors.
Position Vector:
A vector representing the position of a point in space with respect to the origin of a co-ordinate system is known as position vector of the point.
The position vector of point P (x, y, z) is given by
r = x (along x axis) + y (along y axis) + z (along z axis)
or, 
….…. (i)
Magnitude of the position vector is given by 
………. (ii)
Unit vector along the given vector (
)
…. (iii)
Displacement Vector
Let a particle is initially present at point P (x1, y1, z1) and it moves to another point Q (x2, y2, z2) then the position vector of initial and final position are
………. (i)
……… (ii)
Then the displacement vector is given by
EXAMPLE. A particle initially present at point (1m, 2m, 1m) moves to another point (3m, -1m, 2m) in 2 sec. Determine its average velocity.
Solution. (1m, 2m, 1m) = (x1, y1, z1)
(3m, -1m, 2m) = (x2, y2, z2)
Or 
Or 
Or 
Magnitude of the average velocity is
Addition of Vectors
What is vector addition rule?
To add two given vectors and by geometrical method, we redraw the first vector, then from the head of the first vector we draw the second vector and then the vector starting from the tail of the first vector ending at the head of the second vector represents the resultant of addition of the two vectors.
Law of Triangle of vector addition:
According to the law of Triangle of Vector addition
“If two given vectors can be represented both in its magnitude and direction by two sides of a triangle in an order, then the third side of the triangle represents the sum of the vectors (resultant), both in magnitude and direction, when taken in opposite order”.
.
In the fig. A is the first vector and B is the second vector. A and B are represented by two sides of the triangle in the same order where as, resultant R, which a vector represented by third side of the triangle and taken in opposite order.
Law of Parallelogram of vector addition:
According to the law of Parallelogram of Vector addition
“If two vectors acting at the same point can be represented by two adjacent sides of a parallelogram, both in its magnitude and direction, then the diagonal of the diagram starting from the same point represents the sum of the two vectors (resultant), both in is magnitude and direction”.
Law of polygon of vector addition
According to the law of Polygon of Vector addition
“If three or more than three given vectors can be represented both in its magnitude and direction by consecutive sides of an open convex polygon in an order then the closing side of the polygon represents the sum of the vectors (resultant), both in its magnitude and direction, when taken in opposite order”.
In the fig. A is the first vector and B is the second vector. A and B are represented by two consecutive sides of a parallelogram (sides ab and ac) and both are starting from same point. The resultant R, is a vector represented by that diagonal of the parallelogram which is starting from the same point from which the two given vectors started.
In the fig. A, B, C and D are the four given vectors. These vectors are represented one after the other, in a order (anticlockwise), by consecutive sides of a polygon. The closing side of the polygon (side ae) represents the sum (resultant) of vectors when taken in opposite order (clockwise).
Laws of Vector Addition (Video)
(In Hindi + English Mix)
Addition of Two Vectors Formula (Analytical method)
In ∆adc,
ac2 = ad2 + dc2
ac2 = (ab + bd) 2 + dc2
ac2 = ab2 + bd2 + 2.(ab).(bd) + dc2 ……. (i)
ac2 = 
………. (ii)
ab = 
……………(iii)
In ∆bdc
……………………. (iv)
………………….. (v)
Putting these values in equation (i)
……………. (vi)
In ∆bdc
………………………… (vii)
Addition Of Vectors : Analytical Method (Video)
(In Hindi + English Mix)
Example.
Two forces f1 = 3N, E and f2 = 4N, N acts on a body of mass 5kg. Determine
(i) the net force acting on the body
(ii) the acceleration of the body.
Solution.
ac2 = ab2 + bc2
so, the net force is 5N, 530 North of East.
or 
The direction of acceleration is same as that of the net force.
Example. A man moves 100m towards east then he turns 600 to his initial direction towards his left and again moves by 100m. Determine his net displacement.
Solution.
= 100m, E
= 100m, 600 N of E
and θ = 600
= 100 × 1.732m = 173.2m
So, α= 300
Net displacement is 173.2m, 300 North of East.
Subtraction of Vectors
Since, 
or , equivalently 
This indicates that subtraction can be considered equivalent to ” addition of negetive of the second vector to the first vector”.
This indicates that subtraction can be considered equivalent to ” addition of negetive of the second vector to the first vector”.
The magnitude of subtraction of vector is
……………….. [1]
The angle between the resultant and the first vector ‘α’ is given by
……………………………..[2]
Numerical Problems on subtraction of vectors
Example. A motorcyclist moving towards East with speed 20 km/hr takes a sharp turn and then moves towards North keeping his speed unchanged. Determine the change in his velocity.
Solution. Since the speed remains the same
or 
km/hr = 20 × 1.414 km/hr = 28.28 km/hr
Again, from fig 
or α = 450
Thus, the change in velocity of the motorcyclist is 28.28 km/hr, 450 West of North.
Example. Two vectors 
and 
are such that it satisfies the conditions 
. Determine the angle between and .
Solution.
Squaring (i) and (ii)
then cos θ = 0 or θ = 900
Example . Two vectors and of same magnitude when added the resultant is also a vector of same magnitude. Determine the angle between the vectors and .
Solution.
x2 = x2 + x2 + 2.x.x.cos θ
x2 = 2x2 + 2x2.cos θ
1 = 2 + 2 cos θ
cos θ = or cos θ = – cos 600 or cos θ = cos (180 – 60)
θ = 180 – 600 or θ = 1200
Example. Sum of two vectors and is such that it is perpendicular to one of the vector and its magnitude is half of the larger vector. Determine the angle between two given vectors.
Solution.
Consider the figure. It is given that
So, the angle between the vectors = 1800 – α
θ = 1800 – 300 or θ = 1500
