Question 2: Explain in detail whether for two vectors of equal magnitudes is it possible to give a resultant of magnitude equal to their individual magnitudes? Justify your answer mathematically.

Answer

As vector is a directed quantity, therefore, its addition depends upon the direction of the vectors to be added. Hence, it is possible that the resultant of two vectors of equal magnitudes yield a vector of the same magnitude provided we look for an appropriate angle between the two vectors.

Suppose two vectors and are to be added.

Let is acting along the x-axis and is making an angle θ with the x-axis.

Now, in terms of magnitudes only,

A_x = A cos0° and since cos0° = 1, therefore,

A_x = A         ————         (1)

A_y = A sin0° . Since sin0° = 0, therefore,

A_y = 0        ————          (2)

Similarly,
B_x = Bcosθ       ———       (3)
B_y = Bsinθ        — —–       (4)
If ‘R’ is the magnitude of the resultant vector, then,

Now, putting the values of equations (1), (2), (3) and (4) in this equation,

As the magnitude of both vectors is the same, therefore put A = B.

Now we have to think about the given condition, i-e, R = A = B and look for some suitable value of θ. The suitable value of θ should be such that the term (1+cosθ) in the above equation gives a value equal to 1/2. So an angle θ for which cosθ = -1/2 would be the appropriate angle. As cos120^o is -1/2, therefore, the required angle is 120^o.

Now A and B are equal as given in the problem, therefore, for θ = 120⁰ the condition of the problem A = B = R is met.

Thus two vectors of the same magnitudes when added will give a resultant of equal magnitude if the angle between them is 120^o. See the figure.