# Parallelogram Law of Vector Addition

## Statement of Parallelogram Law

If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.

## Derivation of the law

Note: All the letters in bold represent vectors and normal letters represent magnitude only.

Let **P** and **Q** be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure.

Let θ be the angle between **P **and **Q **and **R **be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of **P** and **Q**.

So, we have

**R **=** P **+** Q**

Now, expand A to C and draw BC perpendicular to OC.

From triangle OCB,

In triangle ABC,

Also,

**Magnitude of resultant:**

Substituting value of AC and BC in (i), we get

which is the magnitude of resultant.

**Direction of resultant:** Let ø be the angle made by resultant **R **with **P**. Then,

From triangle OBC,

which is the direction of resultant.

## Numerical Problem

Two forces of magnitude 6N and 10N are inclined at an angle of 60° with each other. Calculate the magnitude of resultant and the angle made by resultant with 6N force.

**Solution:**

Let **P **and** Q** be two forces wih magnitude 6N and 10N respectively and θ be angle between them. Let **R** be the resultant force.

So, P = 6N, Q = 10N and θ = 60°

We have,

which is the required magnitude

Let ø be the angle between **P** and **R**. Then,

which is the required angle.