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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? 

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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.

Find the magnitude and direction angles of the force F3, given that the resultant force FR of the three forces shown is FR = {9j} kN.

Learn how to find the magnitude and direction angles of force F3 in a 3D force equilibrium problem where the resultant force FR is 9 kN in the y-direction. This problem covers vector decomposition, force equilibrium, and angle calculations in a three-dimensional space. Ideal for college-level engineering students.

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