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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