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To determine the direction and magnitude of the reaction at the ceiling where two ropes are attached, let's break down the problem using vector components.

### Step 1: Understanding the Given Forces and Angles

1. **Forces**:
   - \( \vec{F_1} = 500 \, \text{N} \) (at 75° from the ceiling line on the left side)
   - \( \vec{F_2} = 250 \, \text{N} \) (at 45° from the ceiling line on the right side)

2. **Angles**:
   - For \( \vec{F_1} \): 75° from the horizontal
   - For \( \vec{F_2} \): 45° from the horizontal

### Step 2: Resolve Each Force into Horizontal and Vertical Components

For each force \( F \) with angle \( \theta \) from the horizontal:
- **Horizontal component**: \( F_x = F \cos(\theta) \)
- **Vertical component**: \( F_y = F \sin(\theta) \)

#### \( \vec{F_1} = 500 \, \text{N} \), \( \theta_1 = 75^\circ \)
- \( F_{1x} = 500 \cos(75^\circ) \)
- \( F_{1y} = 500 \sin(75^\circ) \)

#### \( \vec{F_2} = 250 \, \text{N} \), \( \theta_2 = 45^\circ \)
- \( F_{2x} = 250 \cos(45^\circ) \)
- \( F_{2y} = 250 \sin(45^\circ) \)

### Step 3: Calculate Resultant Components for Total Force

Since we’re assuming equilibrium, the reaction at the ceiling, \( \vec{R} \), must balance these forces. We sum the horizontal and vertical components:

- **Total Horizontal Force** \( R_x = F_{1x} + F_{2x} \)
- **Total Vertical Force** \( R_y = F_{1y} + F_{2y} \)

### Step 4: Calculate Magnitude and Direction of Reaction

1. **Magnitude of Reaction**: \( R = \sqrt{R_x^2 + R_y^2} \)
2. **Angle of Reaction** with respect to the horizontal: \( \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \)

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### Related Questions
1. How do vector components affect the equilibrium in such force problems?
2. Can we solve for the reaction if one of the forces is unknown?
3. How would the reaction change if the angles were altered?
4. What would happen if there was an additional force acting downward?
5. How does changing the magnitude of one force affect the overall reaction?

**Tip:** Always double-check angles and directions when resolving forces to avoid errors in trigonometric calculations.

Two ropes are attached to a hook in the ceiling, and a man pulls on each rope as indicated in the figure. Determine the direction and magnitude of the reaction at the ceiling.

Learn how to calculate the reaction at the ceiling when two ropes with forces of 500 N and 250 N are attached at specific angles. This problem covers key concepts in vector resolution, equilibrium of forces, and trigonometry, suitable for high school students.

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