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can you answer the question in the picture

To solve this problem, we need to calculate the resultant force (\( \vec{F} \)) from the two given forces \( \vec{F}_1 = 5.85 \, \text{N} \) and \( \vec{F}_2 = 13.0 \, \text{N} \), and then find its magnitude and direction.

### Step 1: Resolve forces into their components
We are given the angles of the forces relative to the positive x-axis:
- \( \vec{F}_1 \) is 5.85 N at an angle of 45° counterclockwise.
- \( \vec{F}_2 \) is 13.0 N at an angle of 30° clockwise.

#### Components of \( \vec{F}_1 \):
- \( F_{1x} = F_1 \cos(45^\circ) = 5.85 \cos(45^\circ) \)
- \( F_{1y} = F_1 \sin(45^\circ) = 5.85 \sin(45^\circ) \)

#### Components of \( \vec{F}_2 \):
- \( F_{2x} = F_2 \cos(-30^\circ) = 13.0 \cos(30^\circ) \)
- \( F_{2y} = F_2 \sin(-30^\circ) = 13.0 \sin(-30^\circ) \)

Note that the angle of \( F_2 \) is negative because it's clockwise from the positive x-axis.

### Step 2: Add the components
- Resultant x-component: \( F_x = F_{1x} + F_{2x} \)
- Resultant y-component: \( F_y = F_{1y} + F_{2y} \)

### Step 3: Calculate magnitude of the resultant force
The magnitude of the resultant force is given by:
\[
F = \sqrt{F_x^2 + F_y^2}
\]

### Step 4: Calculate direction of the resultant force
The direction \( \theta \) counterclockwise from the positive x-axis is:
\[
\theta = \tan^{-1}\left( \frac{F_y}{F_x} \right)
\]

I'll now calculate this.The magnitude of the resultant force \( F \) is approximately **15.58 N**, and its direction is **8.73° clockwise** from the positive x-axis (since the angle is negative, it indicates a clockwise direction).

Would you like further details or clarifications?

Here are some related questions you might find interesting:
1. How do vector components affect the resultant force?
2. What would happen if one of the forces was acting purely along the x-axis?
3. How can the direction of a vector change the magnitude of the resultant?
4. What role do trigonometric functions play in resolving forces?
5. How can the resultant force direction be interpreted geometrically?

**Tip:** When dealing with force vectors, always pay attention to the angle's reference (clockwise or counterclockwise) to correctly resolve the components.

The particle shown below is at rest, where F1 = 5.85 N, and F2 = 13.0 N. Find the magnitude and direction of F.

Learn how to calculate the magnitude and direction of the resultant force from two given forces, F1 = 5.85 N at 45° and F2 = 13.0 N at -30°, using vector addition and trigonometric formulas. Suitable for high school physics and math students.

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