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To solve the problem, we need to find the resultant vector of the given forces using both graphical and analytical methods. Here's a structured breakdown of how to handle both approaches:

### **Step 1: Analyzing the Given Vectors**
Below is the list of given vectors with their respective magnitudes and directions:

1. \(\vec{A} = 5 \, \text{N} \) at \(40^\circ \, \text{NE}\) 
2. \(\vec{B} = 3 \, \text{N} \) (East)
3. \(\vec{C} = 3 \, \text{N} \) at \(50^\circ \, \text{SE}\)
4. \(\vec{D} = 5 \, \text{N} \) (South)
5. \(\vec{E} = 10 \, \text{N} \) at \(30^\circ \, \text{SW}\)
6. \(\vec{F} = 10 \, \text{N} \) at \(30^\circ \, \text{NW}\)
7. \(\vec{G} = 5 \, \text{N} \) (North)
8. \(\vec{H} = 3 \, \text{N} \) at \(50^\circ \, \text{NE}\)
9. \(\vec{I} = 3 \, \text{N} \) (East)

### **Step 2: Analytical Method**
We'll break down each vector into its horizontal (x) and vertical (y) components using trigonometric functions. The formulas for each component are:

- **Horizontal Component (\(x\))**: \( x = F \cdot \cos(\theta) \)
- **Vertical Component (\(y\))**: \( y = F \cdot \sin(\theta) \)

#### 1. Vector Decomposition:
1. **\(\vec{A}\):** 
   - \(x_A = 5 \cdot \cos(40^\circ)\)
   - \(y_A = 5 \cdot \sin(40^\circ)\)
   
2. **\(\vec{B}\):**
   - \(x_B = 3 \) (East, so no angle needed)
   - \(y_B = 0 \)

3. **\(\vec{C}\):**
   - \(x_C = 3 \cdot \cos(50^\circ) \) (positive because it's SE direction)
   - \(y_C = -3 \cdot \sin(50^\circ) \) (negative since it’s downward)

4. **\(\vec{D}\):**
   - \(x_D = 0 \)
   - \(y_D = -5 \) (directly South)

5. **\(\vec{E}\):**
   - \(x_E = -10 \cdot \cos(30^\circ) \) (negative because it's SW direction)
   - \(y_E = -10 \cdot \sin(30^\circ) \)

6. **\(\vec{F}\):**
   - \(x_F = -10 \cdot \cos(30^\circ) \) (negative because it's NW direction)
   - \(y_F = 10 \cdot \sin(30^\circ) \)

7. **\(\vec{G}\):**
   - \(x_G = 0 \)
   - \(y_G = 5 \) (North)

8. **\(\vec{H}\):**
   - \(x_H = 3 \cdot \cos(50^\circ) \) (positive because it's NE direction)
   - \(y_H = 3 \cdot \sin(50^\circ) \)

9. **\(\vec{I}\):**
   - \(x_I = 3 \) (East)
   - \(y_I = 0 \)

#### 2. Sum of Components:
- Sum all x-components to get the resultant x-component (\(R_x\)).
- Sum all y-components to get the resultant y-component (\(R_y\)).

#### 3. Calculate the Resultant Vector:
The magnitude \( R \) and direction \( \theta \) of the resultant vector can be found using:
- \( R = \sqrt{R_x^2 + R_y^2} \)
- \( \theta = \arctan\left(\frac{R_y}{R_x}\right) \)

### **Step 3: Graphical Method**
For the graphical solution, follow these steps:

1. Use graph paper and the scale given: **1 cm = 1 Newton**.
2. Draw each vector sequentially from the endpoint of the previous vector.
3. Use a ruler and protractor for accurate angles.
4. After plotting all vectors, measure the distance from the start to the endpoint to find the magnitude of the resultant vector.
5. Use a protractor to determine the direction of the resultant vector.

### **Next Steps**
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Solve the resultant of the following given vectors by graphical and analytical methods. Show solutions in a full sheet for analytical and use graph paper or bond paper for graphical methods. Scale: 1 cm = 1 Newton.

Learn to find the resultant of multiple vectors using graphical and analytical methods with step-by-step solutions. This includes breaking down vectors into components and using trigonometric functions for precise calculations. Suitable for students in Grades 10-12.

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