Hello Azhar,
Resultant of two forces P and Q is R
R = P + Q
|R| = sqrt(P^2 + Q^2 + 2PQCosθ)
When Q is doubled new resultant R’ is
R’ = P + 2Q
Since it is perpendicular to P, the dot product of R’ and P is zero
(R’).(P) = 0
(P + 2Q).(P) = 0
P^2 + 2PQCosθ = 0
(Dot product of P and P is P^2 and that of P and 2Q is 2PQCosθ)
Cosθ = -P/(2Q)
R = sqrt(P^2 + Q^2 + 2PQCosθ)
R = sqrt(P^2 + Q^2 + 2PQ*(-P/(2Q))
R = sqrt(P^2 + Q^2 - P^2)
R = Q
Hope this helps.
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