Based on the provided image, this problem involves calculating the resultant force location (Part B) after determining the magnitude of the resultant force (Part A, already answered as 26.0 kN). The setup involves forces $F_A = 9 \, \text{kN}$ and $F_B = 3 \, \text{kN}$, acting on a structure shown in the figure with dimensions labeled in millimeters.

To determine the location of the resultant force, you need to calculate the coordinates $(x_R, y_R)$ in the x-y plane using the principle of moments. Here's a general process for finding the centroid or resultant location:

Steps:

1. Determine the Moments: The moment caused by each force about the origin $O$ can be found by multiplying the force by the perpendicular distance from the origin.

   For $F_A$:

   • Position: 700 mm along the x-axis and 100 mm along the y-axis.

   For $F_B$:

   • Position: 150 mm along the x-axis and 750 mm along the y-axis.

2. Calculate Resultant Force: Since the magnitude of the resultant force $F_R$ is already known, we need the location of this resultant in the x and y directions.

   The formula for the x and y coordinates is: $x_R = \frac{\sum (F_i \cdot x_i)}{F_R}$ $y_R = \frac{\sum (F_i \cdot y_i)}{F_R}$

3. Substitute the Values: For $x_R$: $x_R = \frac{(9 \, \text{kN} \times 700 \, \text{mm}) + (3 \, \text{kN} \times 150 \, \text{mm})}{26 \, \text{kN}}$ For $y_R$: $y_R = \frac{(9 \, \text{kN} \times 100 \, \text{mm}) + (3 \, \text{kN} \times 750 \, \text{mm})}{26 \, \text{kN}}$

Let's compute these values.The calculated location of the resultant force in the x-y plane is:

$x_R = 260 \, \text{mm}, \quad y_R = 121 \, \text{mm}$

These values are rounded to three significant figures.

Would you like further details or explanations?

Here are some related questions:

1. How is the moment of a force calculated?
2. How would the result change if the forces were applied in opposite directions?
3. What is the significance of the resultant force in mechanical systems?
4. How do you interpret the moment about a point in physics?
5. Why is it important to specify the exact location of the resultant force?

Tip: Always ensure your coordinate system is defined clearly when solving for forces and moments to avoid errors in sign or direction.