# The Law of Parallelogram of Forces

The **law of parallelogram of forces** states that if two vectors acting on a particle at the same time be represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point .

Okay, got it what it wants to say. But, the main question here is how does it happen! Right?

So let us assume that the two vectors A and B, inclined at angle θ, be acting on a particle at the same time. Let they be represented in magnitude and direction by two adjacent sides OP and OS of parallelogram OPQS, drawn from a point O.

According to parallelogram law of vectors , their resultant vector will be represented by the diagonal of the parallelogram .

**Magnitude and Direction of Resultant:**

Draw a perpendicular QN to OP produced.

And let us assume that OP=A, OS= PQ= B, OQ=R and angle SOP= angle QPN = θ.

Now considering this if we proceed further , in the case of triangle law of vector addition , the magnitude and direction of resultant vector will be given by

R= sqrt of A^2 + B^2 + 2 AB cosθ

tan B = B sinθ/ A+B cosθ

**Special cases :-**

**(1) When two vectors are acting in the same direction , then θ= 0 , cosθ=1 and sinθ= 0 **

R= sqrt of A^2 + B^2 + 2 AB

=sqrt of (A+B)^2 = A + B

tan Beta = B X 0/ A+B = 0

Beta = 0

Thus for two vectors acting in the same direction the magnitude of the resultant vector is equal to the sum of the magnitudes of two vectors and act along the direction of A and B.

(2) **When two vectors are acting in opposite directions , then θ= 180 , cos θ= -1 and sinθ= 0**

R= sqrt of A^2+ B^2+ 2 AB (-1)

= sqrt of (A-B) or (B-A)

tan beta = B X 0/ A+ B (-1)= 0

Beta = 0 or 180.

Thus for two vectors acting in opposite directions, the magnitude of the resultant vector is equal to the difference of the magnitudes of the two vectors and acts in the direction of bigger vector .

(3) **When two vectors act at right angle to each other θ = 90 , sin θ = 1 and cosθ = 0 **

R= sqrt of A^2+B^2 + 2 AB (0)

= sqrt of A^2+B^2

tan beta = B(1)/A+B(0)= B/A

or, Beta = tan^-1 B/A

**IMPORTANT NOTE :**

1. It is to be noted that the magnitude of the resultant of two vectors is maximum , when the vectors act in the same direction and is minimum when they act in opposite directions.

2. It should be noted that while finding the resultant vector of two vectors by the parallelogram law of vector addition , the two vector A and B should be either act towards the point or away from the point .

**Questions based upon parallelogram law of forces–**

**Q 1) Two forces 5 N and 20 N are acting at an angle of 120 degree between them . Find the resultant force in magnitude and direction.**

Solution : Here A = 5 N

B = 20 N ; θ= 120 degree ; R= ?; beta =?

R= sqrt of A^2+ B^2+ 2 AB cosθ

= sqrt of 5^2 + 20^2+ 2 X 5 X 20 cos 120 degree

= sqrt of 325

= 18.03 N

tan beta = 5 sin 120 /20 + 5 cos 120

= 5 sqrt 3 / 2 /20 +5 X (-0.5)

= 0.2475 = tan 13 degree 54 min

hence, beta = 13 degree 54′

**Q 2) .Two forces 10 N and 14 N are acting upon the a body . What can be the maximum and minimum resultant force on the body ?**

Solution : Here, A= 10 N ; B= 14 N

R= sqrt of A^2 + B^2 + 2 AB cosθ

a) R will be maximum if cosθ= 1 , then

R= (A+B)=10+14=24 N in the direction of 14 N

b) R will be minimum if cosθ= -1 ,

then R= (A-B) or (B-A)

= 14- 10

=4 N in the direction of 14 N

**Q 3). State the law of parallelogram with proper diagram.**

Solution : It states that if two vectors acting on a particle at the same time be represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.

**Q 4) When vectors act as maximum and minimum?**

Solution : It is to be noted that the magnitude of the resultant of two vectors is maximum , when the vectors act in the same direction and is minimum when they act in opposite directions.

**Q 5) What we need to consider in this law of parallelogram of forces **

Solution : It should be noted that while finding the resultant vector of two vectors by the parallelogram law of vector addition , the two vector A and B should be either act towards the point or away from the point.

**You may also like to visit:
**Verification of Law of Triangle of Forces and Lami’s Theorem

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I am busy with an Exercise but i am not very familiar with the Parallelogram of forces.

In this exercise the There is one 25 degree angle 75 Newton and 65 newton,so I must get the resultant.

I need help with the formula of getting the Resultant.

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