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State and prove parallelogram law of vectors and discuss the special cases.

 Parallelogram law of vectors

If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal passing through the common tail of the two vectors.

Let us consider two vectors and which are inclined to each other at an angle ?. Let the vectors and be represented in magnitude and direction by the two sides OA and OB of a parallelogram OACB. The diagonal OC passing through the common tail O, gives the magnitude and direction of the resultant .

CD is drawn perpendicular to the extended OA, from C. Let made by with be ?.

From right angled triangle OCD,

 OC² = OD² + CD²

        = (OA + AD)² + CD²

        = OA² + AD² + 2.OA.AD + CD²                 (1)                 

From right angled ? CAD,

AC² = AD² + CD²                                             (2)         

Substituting (2) in (1)

  OC² = OA² + AC² + 2OA.AD                           (3)

From ? ACD,

       CD = AC sin ?                                            (4)

AD = AC cos ?                                                  (5)

Substituting (5) in (3) OC² = OA² + AC² + 2 OA.AC cos ?

Substituting OC = R, OA = P,

OB = AC = Q in the above equation

R² = P² + Q² + 2PQ cos ?

(or)                          (6)

Equation (6) gives the magnitude of the resultant. From ? OCD,

Substituting (4) and (5) in the above equation,

                  (7)

Equation (7) gives the direction of the resultant.

Special Cases

(i) When two vectors act in the same direction

       In this case, the angle between the two vectors ? = 0°, cos 0° = 1, sin 0° = 0

From (6)    

From (7)    

That is ? = 0

Thus, the resultant vector acts in the same direction as the individual vectors and is equal to the sum of the magnitude of the two vectors.

(ii) When two vectors act in the opposite direction

     In this case, the angle between the two vectors ? = 180°, cos 180° = -1, sin 180° = 0.

From (6)  

From (7)  

Thus, the resultant vector has a magnitude equal to the difference in magnitude of the two vectors and acts in the direction of the bigger of the two vectors

(iii) When two vectors are at right angles to each other

      In this case, ? = 90°, cos 90° = 0, sin 90° = 1

From (6)

From (7) 

The resultant vector acts at an angle ? with vector .