A person starts walking from home and walks: 6 miles East 6 miles Southeast 3 miles South 1 miles Southwest 4 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 Question Help:

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Sure, let's tackle each part of the problem step by step.

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

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

- **6 miles East**
- **6 miles Southeast**
- **3 miles South**
- **1 mile Southwest**
- **4 miles East**

**Calculation:**
$$
\text{Total Distance} = 6 + 6 + 3 + 1 + 4 = 20 \text{ miles}
$$

**Rounded to two decimal places:**  
$$
\boxed{20.00 \text{ miles}}
$$

---

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

Displacement is a vector quantity that depends on both **magnitude** and **direction**. To find the total displacement vector, we'll break each movement into its **east (x-axis)** and **north (y-axis)** components, sum them up, and then determine the resultant vector.

**Assumptions:**
- **East** is positive along the **x-axis**.
- **North** is positive along the **y-axis**.

Let's break down each movement:

1. **6 miles East**
   - **x-component:** $$ +6 $$ miles
   - **y-component:** $$ 0 $$ miles

2. **6 miles Southeast**  
   Southeast is **45° South of East**.
   - **x-component:** $$ 6 \times \cos(45^\circ) \approx 6 \times 0.7071 \approx +4.24 $$ miles
   - **y-component:** $$ 6 \times \sin(45^\circ) \approx 6 \times (-0.7071) \approx -4.24 $$ miles *(negative because it's South)*

3. **3 miles South**
   - **x-component:** $$ 0 $$ miles
   - **y-component:** $$ -3 $$ miles

4. **1 mile Southwest**  
   Southwest is **45° South of West**.
   - **x-component:** $$ 1 \times \cos(45^\circ) \approx 1 \times (-0.7071) \approx -0.71 $$ miles
   - **y-component:** $$ 1 \times \sin(45^\circ) \approx 1 \times (-0.7071) \approx -0.71 $$ miles

5. **4 miles East**
   - **x-component:** $$ +4 $$ miles
   - **y-component:** $$ 0 $$ miles

**Summing Components:**

- **Total x-component:** 
  $$
  6 + 4.24 + 0 + (-0.71) + 4 = 13.53 \text{ miles East}
  $$
  
- **Total y-component:**
  $$
  0 + (-4.24) + (-3) + (-0.71) + 0 = -7.95 \text{ miles South}
  $$

**Total Displacement Vector:**  
$$
\boxed{13.54 \text{ miles East and } 7.95 \text{ miles South}}
$$

*(Rounded to two decimal places)*

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### **3. Straight-Line Distance to Walk Home**

To find the **straight-line distance** back home, calculate the **magnitude** of the displacement vector using the Pythagorean theorem.

**Given:**
- **Eastward displacement (x):** $$ 13.54 $$ miles
- **Southward displacement (y):** $$ 7.95 $$ miles

**Calculation:**
$$
\text{Distance} = \sqrt{(13.54)^2 + (7.95)^2}
$$
$$
\text{Distance} = \sqrt{183.07 + 63.20} = \sqrt{246.27} \approx 15.70 \text{ miles}
$$

**Rounded to two decimal places:**  
$$
\boxed{15.70 \text{ miles}}
$$

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### **Summary of Answers**

1. **Total Distance Walked:**  
   $$\boxed{20.00 \text{ miles}}$$

2. **Total Displacement Vector:**  
   $$\boxed{13.54 \text{ miles East and } 7.95 \text{ miles South}}$$

3. **Straight-Line Distance to Walk Home:**  
   $$\boxed{15.70 \text{ miles}}$$

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If you have any further questions or need more clarification on any of the steps, feel free to ask!
The person walked a total of 20.00 miles. The total displacement vector is 13.54 miles East and 7.95 miles South. If they walked straight home, they'd have to walk 15.70 miles.

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