https://r2.erweima.ai/mathbot/image/1726718298141u00msb6w.jpg

Let's begin by analyzing the vectors from the image. There are four vectors to be summed:

1. **Vector A**: Magnitude = 6u, Angle = 50° from the horizontal.
2. **Vector B**: Magnitude = 9u, Angle = 32° from the horizontal.
3. **Vector C**: Magnitude = 5.5u, Angle = 43° from the horizontal.
4. **Vector D**: Magnitude = 5u, Angle = 50° downward from the horizontal.

We will first find the components of each vector and then sum their horizontal and vertical components.

### Step 1: Decompose each vector into horizontal and vertical components.

1. **Vector A:**
   - \( A_x = 6 \cdot \cos(50^\circ) \)
   - \( A_y = 6 \cdot \sin(50^\circ) \)

2. **Vector B:**
   - \( B_x = 9 \cdot \cos(32^\circ) \)
   - \( B_y = 9 \cdot \sin(32^\circ) \)

3. **Vector C:**
   - \( C_x = 5.5 \cdot \cos(43^\circ) \)
   - \( C_y = 5.5 \cdot \sin(43^\circ) \)

4. **Vector D:**
   - \( D_x = 5 \cdot \cos(50^\circ) \) (angle is downward, so component will be negative for vertical)
   - \( D_y = -5 \cdot \sin(50^\circ) \)

### Step 2: Sum the horizontal and vertical components.

The total horizontal component \( R_x \) and vertical component \( R_y \) can be found by summing the individual components:

- \( R_x = A_x + B_x + C_x + D_x \)
- \( R_y = A_y + B_y + C_y + D_y \)

### Step 3: Find the magnitude and direction of the resultant vector.

The resultant magnitude \( R \) and angle \( \theta \) are given by:

- \( R = \sqrt{R_x^2 + R_y^2} \)
- \( \theta = \tan^{-1}\left( \frac{R_y}{R_x} \right) \)

Let me compute the values for you.The resultant vector after summing the vectors \( \mathbf{A} \), \( \mathbf{B} \), \( \mathbf{C} \), and \( \mathbf{D} \) has:

- **Magnitude**: \( 20.90 \, \text{u} \)
- **Direction**: \( 26.38^\circ \) above the horizontal.

Would you like further details on this calculation, or have any specific questions?

Here are some related questions that expand on this:

1. How are vector components used in real-life engineering problems?
2. What are alternative methods for vector addition besides component-wise addition?
3. How can we visually represent vector addition using the parallelogram method?
4. How does the direction of a vector change if we invert its components?
5. How would this problem change if one of the angles was negative?

**Tip**: When adding vectors graphically, the **parallelogram rule** provides an intuitive visual representation of the resultant.

Suma los vectores \( \mathbf{A} \), \( \mathbf{B} \), \( \mathbf{C} \), y \( \mathbf{D} \). Luego, usa el método del paralelogramo para sumar los vectores \( \mathbf{A} \) y \( \mathbf{B} \).

Learn how to sum vectors \( \mathbf{A} \), \( \mathbf{B} \), \( \mathbf{C} \), and \( \mathbf{D} \) by decomposing them into components and using trigonometry. Also, explore the parallelogram method for vector addition. Suitable for students in Grades 11-12.

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