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To find the resultant force of the three forces in the image, we need to resolve each force into its horizontal (\( x \)-component) and vertical (\( y \)-component), and then sum up the components to find the net force in both directions.

### Given dataTo find the resultant force of the three forces in the image, we need to resolve each force into its horizontal (\( x \)-component) and vertical (\( y \)-component), and then sum up the components to find the net force in both directions.

### Given 
- \( F_1 = 2 \, \text{N} \) at \( 45^\circ \)
- \( F_2 = 4 \, \text{N} \) at \( 60^\circ \)
- \( F_3 = 10 \, \text{N} \) at \( -30^\circ \)

### Step 1: Resolve each force into components.

1. **Force \( F_1 \):**
   - \( F_{1x} = F_1 \cos(45^\circ) = 2 \cos(45^\circ) \)
   - \( F_{1y} = F_1 \sin(45^\circ) = 2 \sin(45^\circ) \)

2. **Force \( F_2 \):**
   - \( F_{2x} = F_2 \cos(60^\circ) = 4 \cos(60^\circ) \)
   - \( F_{2y} = F_2 \sin(60^\circ) = 4 \sin(60^\circ) \)

3. **Force \( F_3 \):**
   - \( F_{3x} = F_3 \cos(-30^\circ) = 10 \cos(-30^\circ) \)
   - \( F_{3y} = F_3 \sin(-30^\circ) = 10 \sin(-30^\circ) \)

### Step 2: Calculate the components.

I will now calculate the components of each force.The components of the resultant force are:

- \( F_x = 12.07 \, \text{N} \)
- \( F_y = -0.12 \, \text{N} \)

The resultant force has a magnitude of approximately \( 12.08 \, \text{N} \), and it makes an angle of \( -0.58^\circ \) with the positive \( X \)-axis, meaning it is almost aligned with the horizontal axis, slightly pointing downwards.

Would you like further details on any part of this calculation, or do you have any additional questions?

Here are five related questions to consider:
1. How would the resultant change if one of the forces was doubled?
2. What is the physical significance of the negative angle of the resultant?
3. Can you find the equilibrium force that would balance these three forces?
4. How does changing the angle of \( F_3 \) affect the overall resultant?
5. What would happen if all forces were applied at \( 90^\circ \) intervals?

**Tip:** Always break forces into components when dealing with non-parallel vectors for easier calculation of the resultant.

Learn how to calculate the resultant force from multiple vectors by resolving them into their x and y components using trigonometric functions. This step-by-step guide covers forces of 2 N, 4 N, and 10 N at different angles, showing the calculation of the resultant magnitude and direction.

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