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The problem you're asking about involves the resultant of four vectors. These vectors act on the same point and have the following magnitudes and directions:

1. **Vector \( \vec{V}_1 \) = 70 kgf** at an angle of 0° (along the positive x-axis).
2. **Vector \( \vec{V}_2 \) = 90 kgf** at an angle of 45°.
3. **Vector \( \vec{V}_3 \) = 100 kgf** at an angle of 30° above the negative x-axis.
4. **Vector \( \vec{V}_4 \) = 70 kgf** at an angle of 20° below the negative x-axis.

### Step-by-Step Solution (Analytical)

We will solve this using the **component method** by breaking each vector into its horizontal (\( x \)) and vertical (\( y \)) components. Then, we'll sum the components to find the resultant vector.

#### 1. **Vector \( \vec{V}_1 \)**
- Magnitude: 70 kgf
- Angle: 0° (purely horizontal)

\[
V_{1x} = 70 \cos(0°) = 70 \text{ kgf}, \quad V_{1y} = 70 \sin(0°) = 0 \text{ kgf}
\]

#### 2. **Vector \( \vec{V}_2 \)**
- Magnitude: 90 kgf
- Angle: 45°

\[
V_{2x} = 90 \cos(45°) = 90 \times \frac{1}{\sqrt{2}} = 63.64 \text{ kgf}
\]
\[
V_{2y} = 90 \sin(45°) = 90 \times \frac{1}{\sqrt{2}} = 63.64 \text{ kgf}
\]

#### 3. **Vector \( \vec{V}_3 \)**
- Magnitude: 100 kgf
- Angle: 30° above the negative x-axis (angle measured from the negative x-axis, so direction is \(-180° + 30° = 150°\))

\[
V_{3x} = 100 \cos(150°) = 100 \times (-\frac{\sqrt{3}}{2}) = -86.60 \text{ kgf}
\]
\[
V_{3y} = 100 \sin(150°) = 100 \times \frac{1}{2} = 50 \text{ kgf}
\]

#### 4. **Vector \( \vec{V}_4 \)**
- Magnitude: 70 kgf
- Angle: 20° below the negative x-axis (\(-180° - 20° = 200°\))

\[
V_{4x} = 70 \cos(200°) = 70 \times (-\cos(20°)) = -65.78 \text{ kgf}
\]
\[
V_{4y} = 70 \sin(200°) = 70 \times (-\sin(20°)) = -23.94 \text{ kgf}
\]

### Total Components
Now, sum the x-components and y-components:

#### Total \( x \)-component:
\[
R_x = V_{1x} + V_{2x} + V_{3x} + V_{4x}
\]
\[
R_x = 70 + 63.64 - 86.60 - 65.78 = -18.74 \text{ kgf}
\]

#### Total \( y \)-component:
\[
R_y = V_{1y} + V_{2y} + V_{3y} + V_{4y}
\]
\[
R_y = 0 + 63.64 + 50 - 23.94 = 89.70 \text{ kgf}
\]

### Resultant Vector
The magnitude of the resultant vector is given by:

\[
R = \sqrt{R_x^2 + R_y^2}
\]
\[
R = \sqrt{(-18.74)^2 + (89.70)^2} = \sqrt{351.26 + 8047.21} = \sqrt{8398.47} = 91.64 \text{ kgf}
\]

### Direction of the Resultant
The direction (angle \( \theta \)) is given by:

\[
\theta = \tan^{-1} \left( \frac{R_y}{R_x} \right)
\]
\[
\theta = \tan^{-1} \left( \frac{89.70}{-18.74} \right) = \tan^{-1} (-4.79) \approx -78.27°
\]

Since the resultant is in the second quadrant, the final angle is \( 180° - 78.27° = 101.73° \).

### Final Result:
- **Magnitude of the resultant vector**: 91.64 kgf
- **Direction**: 101.73° (measured counterclockwise from the positive x-axis)

Would you like further details on any part of this solution?

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Here are five related questions to expand your understanding:

1. How do you calculate vector components for any given angle?
2. What is the significance of breaking vectors into components?
3. How do the signs of the components help in determining the vector's quadrant?
4. Can you solve a vector problem using the graphical method, and how does it compare to the analytical method?
5. How does the resultant direction change if all vectors were in the first quadrant?

**Tip:** Always double-check your angles, especially when vectors are in quadrants other than the first!

Los siguientes vectores actúan todos sobre un mismo punto. Hallar gráficamente y analíticamente su resultante y su dirección.

Learn how to find the resultant magnitude and direction of multiple vectors acting on a single point using graphical and analytical methods. This example demonstrates vector decomposition, component summation, and resultant vector calculation.

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