Question

State parallelogram law of vector addition.. Find the magnitude and direction of the resultant vector p and q inclined at an angle (theta) with each other what happens when theta=0 and theta =90 degree

Answer 1

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

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.