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The image shows a problem that involves finding the resultant force of three vectors given in terms of their magnitudes and directions in a 2D plane. Specifically, there are three forces with magnitudes:

- \( 3 \, \text{kN} \) acting at \( 65^\circ \) above the positive \( x \)-axis,
- \( 8 \, \text{kN} \) acting along the negative \( x \)-axis,
- \( 5 \, \text{kN} \) acting at \( 30^\circ \) below the positive \( x \)-axis.

### To solve for the resultant force:
1. **Break each force into its components** along the \( x \)- and \( y \)-axes using trigonometry:
   - For force \( F_1 = 3 \, \text{kN} \) at \( 65^\circ \):
     - \( F_{1x} = 3 \cos(65^\circ) \)
     - \( F_{1y} = 3 \sin(65^\circ) \)
   - For force \( F_2 = 8 \, \text{kN} \) along the negative \( x \)-axis:
     - \( F_{2x} = -8 \, \text{kN} \)
     - \( F_{2y} = 0 \, \text{kN} \)
   - For force \( F_3 = 5 \, \text{kN} \) at \( 30^\circ \) below the positive \( x \)-axis:
     - \( F_{3x} = 5 \cos(30^\circ) \)
     - \( F_{3y} = -5 \sin(30^\circ) \)

2. **Sum the components** of all forces:
   - Total \( x \)-component: \( F_x = F_{1x} + F_{2x} + F_{3x} \)
   - Total \( y \)-component: \( F_y = F_{1y} + F_{3y} \)

3. **Find the magnitude of the resultant force**:
   - \( F_{\text{resultant}} = \sqrt{F_x^2 + F_y^2} \)

4. **Determine the angle of the resultant force** with respect to the \( x \)-axis:
   - \( \theta = \tan^{-1}\left( \frac{F_y}{F_x} \right) \)

Would you like me to calculate the components, the resultant force, and its angle explicitly?

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**Relative Questions:**
1. How are force vectors resolved into components in 2D?
2. What is the significance of using trigonometric functions to resolve vectors?
3. How does the direction of a vector impact its component signs?
4. Can you apply these principles to three-dimensional force systems?
5. What is the difference between a scalar and a vector in physics?

**Tip:** When working with vectors, always ensure that angles are measured with respect to a clearly defined reference axis (like the \( x \)-axis) to avoid errors in trigonometric calculations.

For the three forces shown in Figure P4-3, find the resultant force, its magnitude, and direction with respect to the x-axis.

Learn how to find the resultant force of three vectors in 2D using vector addition and trigonometric functions. This solution provides a detailed step-by-step process for resolving forces, finding the resultant force's magnitude, and determining its direction with respect to the x-axis.

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