In this situation, the directions of all four forces are specified by the angles in the free-body diagram, and the magnitude of the weight is known. The three unknowns are the magnitudes of forces \(\vec{A}\text{,}\) \(\vec{B}\text{,}\) and \(\vec{C}\text{.}\)
\begin{equation*}
W = m g = \kg{20}\ \aSI{9.81} = \N{196.2}
\end{equation*}
We will first find unit vectors in the directions of the four forces by inspection of the free-body diagram. This step requires visualizing the component’s unit vectors and determining the angles each makes with the coordinate axis.
\begin{align*}
\hat{\vec{W}} \amp = \langle 0, -1, 0 \rangle \\
\hat{\vec{A}} \amp = \langle \cos \ang{35}, \cos{55}, 0 \rangle\\
\hat{\vec{B}} \amp = \langle -\cos \ang{15} \cos \ang{30}, \cos \ang{75}, -\cos \ang{15} \cos \ang{60} \rangle\\
\hat{\vec{C}} \amp = \langle 0, \cos{70}, \cos \ang{20} \rangle
\end{align*}
Particle equilibrium requires that \(\sum \vec{F} = 0\text{.}\)
\begin{equation*}
A\ \hat{\vec{A}} + B\ \hat{\vec{B}} + C\ \hat{\vec{C}} = - W\ \hat{\vec{W}}
\end{equation*}
This is a 3 \(\times\) 3 system of three simultaneous equations, one for each coordinate direction, which needs to be solved for \(A\text{,}\) \(B\text{,}\) and \(C\text{.}\)
\begin{align*}
A \cos \ang{35} - B \cos \ang{15} \cos \ang{30} + 0 \amp = 0 \amp \amp \amp (\Sigma F_x \amp= 0)\\
A \cos \ang{55} + B \cos \ang{75} + C \cos \ang{70} \amp = \N{196.2} \amp \amp \amp (\Sigma F_y \amp= 0)\\
0 - \cos \ang{15} \cos \ang{60} + C \cos \ang{20} \amp = 0 \amp \amp \amp (\Sigma F_z \amp= 0)
\end{align*}
These can be solved by any method you choose. Here we will use Sage. Evaluating the coefficients and expressing the equations in matrix form gives
\begin{equation*}
\begin{bmatrix} 0.819 \amp -0.837 \amp 0 \\ 0.574 \amp 0.259 \amp 0.342 \\ 0 \amp -0.482 \amp 0.940 \end{bmatrix} \begin{bmatrix} A \\ B \\ C\\ \end{bmatrix} = \begin{bmatrix} 0 \\ \N{196.2} \\ 0 \end{bmatrix}\text{.}
\end{equation*}
This is an equation in the form
\begin{equation*}
[A][x] =[B]\text{.}
\end{equation*}
Entering the coefficient matrices into Sage.
After evaluating, we learn that
\begin{align*}
A \amp = \N{196.4} \amp B \amp = \N{192.2} \amp C \amp = \N{98.5}
\end{align*}
.