Answer to Question #290388 in Physics for AMY

Question #290388

The displacement vectors A and B, when added together, give the resultant vector R, so that R = A + B. Use the data in the drawing and the fact that φ = 31° to find the magnitude R of the resultant vector and the angle θ that it makes with the +x axis.


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Expert's answer
2022-01-25T06:50:14-0500

To find the magnitude R of the resultant vector, we need three things: the magnitude A|\vec A| of A, the magnitude B|\vec B| of B, and the angle ϕ\phi between them:


Ax=A,Bx=Bcosϕ,Ay=0,By=Bsinϕ. Rx=Ax+Bx=A+Bcosϕ,Ry=Ay+By=Bsinϕ, R=(Rx)2+(Ry)2.A_x=|\vec A|,\\ B_x=|\vec B|\cos\phi,\\ A_y=0,\\ B_y=|\vec B|\sin\phi. \\\space\\ R_x=A_x+B_x=|\vec A|+|\vec B|\cos\phi,\\ R_y=A_y+B_y=|\vec B|\sin\phi,\\\space\\ |\vec R|=\sqrt{(R_x)^2+(R_y)^2}.


To find the angle θ\theta of R above the x-axis:


θ=arctanRyRx=arctanBsinϕ,A+Bcosϕ.\theta=\arctan\dfrac{R_y}{R_x}=\arctan\dfrac{|\vec B|\sin\phi,}{|\vec A|+|\vec B|\cos\phi}.

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