Determine the magnitudes of F_1, F_2, and F_3 for equilibrium of the particle.

#### Solution:

This question involves expressing forces in Cartesian vector form. If you are unsure on how to do this, read the detailed guide on expressing forcesÂ in Cartesian notation.

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We will first write each force in Cartesian vector form.

F_1\,=\,\left\{F_1\cos60^0i+0j+F_1\sin60^0k\right\}\,=\,\left\{0.5F_1i+0j+0.87F_1k\right\}

F_2\,=\,\left\{\dfrac{3}{5}F_2i-\dfrac{4}{5}F_2j+0k\right\}\,=\,\left\{0.6F_2i-0.8F_2j+0k\right\}

F_3\,=\,\left\{-F_3\cos^30^0i-F_3\sin30^0j+0k\right\}\,=\,\left\{-0.87F_3i-0.5F_3j+0k\right\}

F\,=\,\left\{0i+800\sin30^0j-800\cos30^0k\right\}\,=\,\left\{0i+400j-693k\right\}

F_2\,=\,\left\{\dfrac{3}{5}F_2i-\dfrac{4}{5}F_2j+0k\right\}\,=\,\left\{0.6F_2i-0.8F_2j+0k\right\}

F_3\,=\,\left\{-F_3\cos^30^0i-F_3\sin30^0j+0k\right\}\,=\,\left\{-0.87F_3i-0.5F_3j+0k\right\}

F\,=\,\left\{0i+800\sin30^0j-800\cos30^0k\right\}\,=\,\left\{0i+400j-693k\right\}

Now we can write our equation of equilibrium. All forces added together must equal zero because the system is in equilibrium.

\sum \text{F}\,=\,0

F_1+F_2+F_3+F\,=\,0

F_1+F_2+F_3+F\,=\,0

As each force added together must equal zero, each individual component (x, y, z-components) added together must also equal zero.

x-components:

0.5F_1+0.6F_2-0.87F_3\,=\,0

y-components:

-0.8F_2-0.5F_3+400\,=\,0

z-components:

0.87F_1-693\,=\,0

Solving the three equations gives us:

F_1\,=\,796 N

F_2\,=\,147 N

F_3\,=\,564 N