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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.

lq2p11-3

Suppose two vectors Vector Aand Vector Bare to be added.

Let Vector Ais acting along the x-axis and Vector Bis making an angle θ with the x-axis.

Now, in terms of magnitudes only,

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

Ax = A         ————         (1)

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

Ay = 0        ————          (2)

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

right angle triange magnitude

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

ch2q2-1

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

remained

remained2

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 cos120o is -1/2, therefore, the required angle is 120o.

ch2q2-2

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 120o. See the figure.

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