From the images, it appears that this problem involves finding the magnitude and direction angles of the force $F_3$, given that the resultant force $F_R$ of the three forces shown is $F_R = \{9j\} \, \text{kN}$. Here’s a breakdown of how to approach this problem:

Given Data:

1. Resultant force $F_R = 9 \, \text{kN}$ in the $j$ (or $y$) direction.
2. Other forces:
   • $F_1 = 12 \, \text{kN}$ with an angle of 30° in the $x$-$y$ plane.
   • $F_2 = 10 \, \text{kN}$, oriented along some angle relative to the $x$, $y$, and $z$ axes.

Steps to Solve:

1. Resolve Forces $F_1$ and $F_2$:

   • Calculate the components of $F_1$ and $F_2$ in the $x$, $y$, and $z$ directions based on their given angles.

2. Set up Force Equilibrium:

   • Since the resultant force is $9 \, \text{kN}$ in the $j$-direction, all components in the $i$ and $k$ directions must cancel out, allowing us to solve for $F_3$.

3. Determine Magnitude and Direction Angles of $F_3$:

   • After finding the necessary components for equilibrium, calculate the magnitude of $F_3$ and its direction angles $\alpha_3$, $\beta_3$, and $\gamma_3$ with respect to the $x$, $y$, and $z$ axes.

Would you like me to proceed with calculations, or clarify any particular step?

Here are five related questions to further your understanding:

1. How do we determine the components of a force vector in 3D space?
2. What is the significance of each directional angle ($\alpha$, $\beta$, $\gamma$) in vector analysis?
3. How does vector addition help in finding the resultant force?
4. Why does the problem specify a resultant force only in the $j$-direction?
5. What are common applications of force equilibrium analysis in engineering?

Tip: Always double-check vector component directions to avoid sign errors in force equilibrium problems.