# Chapter 2 - Force Vectors - Section 2.3 - Vector Addition of Forces - Problems - Page 30: 10

$F=980lb$ $\phi=19.4^{\circ}$

#### Work Step by Step

We are asked to determine the magnitude of the resultant force, $F_R$, and its direction, measured counterclockwise from the positive x axis. We can apply the law of cosines to find the magnitude of $F_R$. $c=\sqrt{a^2+b^2-2*a*b*\cos(C)}$ $c=\sqrt{800^2+500^2-2*800*500*\cos(95^{\circ})}$ $c=980lb$ $F=980lb$ Using this result, we can apply the law of sines to find the angle between the top rope and $F_R$, ($\theta$) $\sin\theta / 500lb = \sin 95^{\circ} / 980lb$ Solving for $\theta$, we obtain: $\theta=30.6^{\circ}$ We will need to subtract this answer from the given angle in figure, (50^{\circ}), to get the angle to $F_R$ as measured from the x-axis. $\phi=50^{\circ}-30.6^{\circ}=19.4^{\circ}$

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