Question

two forces acting at a point are such that if the direction of one is reversed , the direction of resultant is turned through a right angle. prove that the forces must be equal in magnitude

Answer 1

Let  angle between them is θ and resultant of both angle(R1) make angle α with Q Then,tan(α)=Psin(θ)Q+Pcos(θ)−−−(1)Now,direction of P get reverse angle between Q and reversed P vectoris(1800−θ). Let new resultant R2 make angleβ withQ. tan(β)=P sin(1800−θ)Q+P cos(1800−θ)=P sin(θ)Q−P cos(θ)
(2)Accordingtoquestion,α+β=90.
tan(α+β)=tan(90)
tan(α) +tan(β)/ 1−tan(α)×tan(β)=1/0 tan(α)×tan(β)=1
Putting values from equation(1)and(2)
(Psin(θ)/(Q+Pcos(θ))(Psin(θ)/(Q−Pcos(θ))=1
P^2 sin^2(θ)=Q^2−P^2 cos^2(θ)
P^2(sin^2(θ)+cos^2(θ))=Q^2
P^2=Q^2
P=Q