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use the following resultant formula
R = [A 2 + B 2 + 2ABCosθ] 1/2
where 'A 'and 'B' are two forces inclined at an angle 'θ' to each other giving a resultant force 'R'.
in the first case case
R1 = p
A = p
and B = q
so,
p2 = p2 + q2 + 2pqCosθ
or
q2 + 2pqCosθ = q + 2pCosθ = 0 (1)
or in other terms
-q2 = 2pqCosθ (2)
and in the second case
A = 2p
B = q
thus,
R22 = 4p2 + q2 + 4pqCosθ = 4p2 + q2 +2(2pqCosθ)
by using equation (1)
R22 = 4p2 + q2 - 2q2
or the new resultant is
R2 = [4p 2 - q 2 ] 1/2 (3)
the angle between R2 and q would be
α = tan-1[qSinθ / (2p + qCosθ)]
according to (1),
2p + qCosθ = 0
thus
tanα = qSinθ / 0 = undefined
α = 90 o
so, the forces q and R2 are perpendicular to each other.
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