A person starts walking from home and walks: 5 miles East 2 miles Southeast 6 miles South 4 miles Southwest 3 miles East Round your answers to 2 decimal places. This person has walked a total of Find the total displacement vector for this walk: If this person walked straight home, they'd have to walk Submit Question

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Let's break down the problem step by step:

### 1. **Total Distance Walked**

To find the total distance walked, simply add up all the individual distances:

- **5 miles East**
- **2 miles Southeast**
- **6 miles South**
- **4 miles Southwest**
- **3 miles East**

**Total Distance = 5 + 2 + 6 + 4 + 3 = 20 miles**

**Rounded to two decimal places:**  
**Total Distance Walked = 20.00 miles**

---

### 2. **Total Displacement Vector**

Displacement considers both magnitude and direction. We'll break each segment into its eastward (x) and northward (y) components. Assume:
- **East** is the positive x-direction.
- **North** is the positive y-direction.
- **South** and **West** are negative directions.

**Breaking Down Each Segment:**

1. **5 miles East**
   - Δx = +5
   - Δy = 0

2. **2 miles Southeast (45° below East)**
   - Δx = $$2 \times \cos(45°) = 2 \times \frac{\sqrt{2}}{2} ≈ +1.41$$
   - Δy = $$2 \times \sin(45°) = 2 \times \frac{\sqrt{2}}{2} ≈ -1.41$$ (South)

3. **6 miles South**
   - Δx = 0
   - Δy = -6

4. **4 miles Southwest (45° below West)**
   - Δx = $$4 \times \cos(225°) = 4 \times \left(-\frac{\sqrt{2}}{2}\right) ≈ -2.83$$
   - Δy = $$4 \times \sin(225°) = 4 \times \left(-\frac{\sqrt{2}}{2}\right) ≈ -2.83$$

5. **3 miles East**
   - Δx = +3
   - Δy = 0

**Calculating Total Displacement:**

- **Total Δx = 5 + 1.41 - 2.83 + 3 ≈ 6.58 miles East**
- **Total Δy = 0 - 1.41 - 6 - 2.83 ≈ -10.24 miles South**

**Total Displacement Vector:**  
$$
\vec{D} \approx (6.58\, \text{East},\ 10.24\, \text{South})
$$

**Rounded to two decimal places:**  
$$
\vec{D} \approx (6.59\, \text{miles East},\ 10.24\, \text{miles South})
$$

---

### 3. **Straight-Line Distance Back Home**

This is the magnitude of the displacement vector, calculated using the Pythagorean theorem:

$$
\text{Distance} = \sqrt{(6.59)^2 + (10.24)^2} ≈ \sqrt{43.35 + 104.90} ≈ \sqrt{148.25} ≈ 12.18\, \text{miles}
$$

**Rounded to two decimal places:**  
**Straight-Line Distance Back Home = 12.18 miles**

---

### **Summary**

- **Total Distance Walked:** 20.00 miles
- **Total Displacement Vector:** 6.59 miles East and 10.24 miles South
- **Straight-Line Distance Back Home:** 12.18 miles
**Total Distance Walked:** 20.00 miles **Total Displacement Vector:** 6.59 miles East and 10.24 miles South **Straight-Line Distance Back Home:** 12.18 miles

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