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 and and be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.

So, we have

                  = 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 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.