Vector components

The force R = 12 N is divided into two components, F1 and F2. Their directions make angles α = 30°, β = 45° with the direction R. What are the components F1 and F2?

Correct answer:

F₁ =  6.2117 N
F₂ =  8.7846 N

Step-by-step explanation:

$$\begin{gathered} R=12\text{N} \\ \alpha =30{}^{\circ } \\ \beta =45{}^{\circ } \\ x=R\cdot \cos \beta =R\cdot \cos 45°=12\cdot \cos 45°=12\cdot 0.707107=8.48528 \\ y=R\cdot \sin \beta =R\cdot \sin 45°=12\cdot \sin 45°=12\cdot 0.707107=8.48528 \\ \gamma =\alpha +\beta =30+45=75{}^{\circ } \\ x=F_1+F_2\cdot \cos \gamma \\ y=0+F_2\cdot \sin \gamma \\ {F}_2=y/\sin \gamma =y/\sin 75°=8.4853/\sin 75°=8.4853/0.965926=8.78461\text{N} \\ {F}_1=x-F_2\cdot \cos \gamma =x-F_2\cdot \cos 75°=8.4853-8.7846\cdot \cos 75°=8.4853-8.7846\cdot 0.258819=6.212=6.2117\text{N} \end{gathered}$$

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