# Law of Parallelogram of Forces

## Law of Parallelogram of Forces

Law of Parallelogram of Forces states that if two forces acting simultaneously on a body at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces.

### Formula for Magnitude of Resultant Force R

Where R is the resultant force and θ is the angle between P and Q.

### Formula for Direction of Resultant Force R

## Cases for Parallelogram Law of Vector Addition

### 1. When two vectors are acting in the same direction , then θ = 0 and cosθ = 1

When two vectors are parallel and have the same direction, the law simplifies to simple addition then the resultant vector will have the same direction as the original vectors and a magnitude equal to the sum of their magnitudes.

R^{2} = P^{2} + Q^{2} + 2PQ [As cosθ = 1]

or, R^{2} = (P + Q)^{2}

or, R = P + Q

Thus, when two vectors are in the same direction, the resultant vector is their algebraic sum given by using R = P + Q where R is the resultant vector, P and Q are the parallel vectors.

### 2. When two vectors are acting in opposite directions , then θ = 180 and cos θ = –1

When two vectors are in opposite directions, the resultant vector will have a magnitude of zero because the vectors cancel each other out.

R^{2} = P^{2} + Q^{2} – 2PQ [As cosθ = –1]

or, R^{2} = (P – Q)^{2}

or, R = P – Q

### 3. When two vectors act at right angle to each other θ = 90 and cosθ = 0

When two vectors are perpendicular to each other, by using the Pythagorean theorem, we can find magnitude of the resultant vector is equal to the square root of the sum of the squares of the magnitudes of the two vectors.

R^{2} = P^{2} + Q^{2} [As cosθ = 0]

We have seen that in the above cases, 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.

## Limitations of Parallelogram Law of Forces

Parallelogram law of forces can only be used when the two forces can form adjacent sides of a parallelogram. If the two forces are parallel to each other or opposite to each other, then the parallelogram law does not work. In other words, it will not work if the angle between the two forces is 0° or 180°.

If the forces are parallel to each other (an angle of 0°), then the forces need to be added together.

### Parallelogram law of forces states that if two forces acting simultaneously at a point be represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant may be represented in magnitude and direction by

A. longer side of the other two sides

B. shorter side of the other two sides

C. diagonal of the parallelogram which does not pass through their point of intersection

D. diagonal for the parallelogram which passes through their point of intersection

## Answer

Option D is correct answer.

diagonal for the parallelogram which passes through their point of intersection. See statement of the Law of Parallelogram of Forces.