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”.
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.
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, 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
(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)
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 2. 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.