two forces f1 and f2 acting on a body have a resultant r1 if f2 is doubled the new resultant r2 forms a right angle with f1.prove that r1 and f2 have same magnitude.

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Solution

Let Î¸ be the angle between f₁ and f₂
(Quantities in bold represents vectors)
Given,
f₁ + f₂= r₁ ----- 1
f₁ + 2f₂ = r₂----2
Taking dot product on itself of eqn. 1
r₁. r₁ = f₁.f₁ +f₁.f₂ + f₂.f₁ + f₂.f₂
=> r₁² = f₁² + f₂² + 2f₁f₂cosÎ¸ ---3.
Taking dot product of f₁ with eqn. 2
f₁.(f₁ + 2f₂) = f₁.r₂
Since, f₁ and r₂ are perpendicular to each other f₁.r₂ = 0
=> f₁² + 2f₁f₂cosÎ¸ = 0 ---4.
By eqn. 3 and 4.
r₁² = (f₁² + 2f₁f₂cosÎ¸) + f₂²
=> r₁² = 0 + f₂²
=> r₁ = f₂
QED.

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