Answer to Question #244559 in Physics for Asmar

Question #244559

Two vectors A and B have precisely equal magnitudes. In order for the magnitude of A + B to be one 

hundred times larger than the magnitude of A - B, what must be the angle between them?


1
Expert's answer
2021-10-02T14:56:10-0400

The magnitude of A+B:


R1=(A+Bcosθ)2+(0+Bsinθ)2==(A+Bcosθ)2+B2sin2θ.R_1=\sqrt{(A+B\cos\theta)^2+(0+B\sin\theta)^2}=\\ =\sqrt{(A+B\cos\theta)^2+B^2\sin^2\theta}.

The Magnitude of A-B:


R2=(ABcosθ)2+(0Bsinθ)2==(ABcosθ)2+B2sin2θ.R_2=\sqrt{(A-B\cos\theta)^2+(0-B\sin\theta)^2}=\\ =\sqrt{(A-B\cos\theta)^2+B^2\sin^2\theta}.

The ratio:


R1R2=100, 100=(A+Bcosθ)2+B2sin2θ(ABcosθ)2+B2sin2θ, 104=(A+Bcosθ)2+B2sin2θ(ABcosθ)2+B2sin2θ, 104=A2+B2cos2θ+2ABcosθ+B2B2cos2θA2+B2cos2θ2ABcosθ+B2B2cos2θ, 104=A2+2ABcosθ+B2A22ABcosθ+B2.\frac{R_1}{R_2}=100,\\\space\\ 100=\frac{\sqrt{(A+B\cos\theta)^2+B^2\sin^2\theta}}{\sqrt{(A-B\cos\theta)^2+B^2\sin^2\theta}},\\\space\\ 10^4=\frac{(A+B\cos\theta)^2+B^2\sin^2\theta}{(A-B\cos\theta)^2+B^2\sin^2\theta},\\\space\\ 10^4=\frac{A^2+B^2\cos^2\theta+2AB\cos\theta+B^2-B^2\cos^2\theta}{A^2+B^2\cos^2\theta-2AB\cos\theta+B^2-B^2\cos^2\theta},\\\space\\ 10^4=\frac{A^2+2AB\cos\theta+B^2}{A^2-2AB\cos\theta+B^2}.

Since A and B have "precisely equal magnitudes," then, putting A=B=V:


104=2V2+2V2cosθ2V22V2cosθ, 104104cosθ=1+cosθ, cosθ=1041104+1, θ=arccos1041104+1=1.15°.10^4=\frac{2V^2+2V^2\cos\theta}{2V^2-2V^2\cos\theta},\\\space\\ 10^4-10^4\cos\theta=1+\cos\theta,\\\space\\ \cos\theta=\frac{10^4-1}{10^4+1},\\\space\\ \theta=\arccos\frac{10^4-1}{10^4+1}=1.15°.


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