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To find the magnitude of the net displacement for the day, we need to add the morning and afternoon displacements vectorially and then calculate the magnitude of the resulting net displacement vector.

### Given Information:
1. **Morning Displacement (\(\vec{S}_{\text{morning}}\)):**
   \[
   \vec{S}_{\text{morning}} = (2200 \, \text{m, east}) + (3000 \, \text{m, north}) + (200 \, \text{m, vertical})
   \]

2. **Afternoon Displacement (\(\vec{S}_{\text{afternoon}}\)):**
   \[
   \vec{S}_{\text{afternoon}} = (1000 \, \text{m, west}) + (2700 \, \text{m, north}) - (500 \, \text{m, vertical})
   \]

### Calculating Net Displacement:

1. **Net Displacement in East-West Direction:**
   \[
   \text{Net East-West} = 2200 \, \text{m} \, (\text{east}) - 1000 \, \text{m} \, (\text{west}) = 1200 \, \text{m} \, (\text{east})
   \]

2. **Net Displacement in North-South Direction:**
   \[
   \text{Net North-South} = 3000 \, \text{m} \, (\text{north}) + 2700 \, \text{m} \, (\text{north}) = 5700 \, \text{m} \, (\text{north})
   \]

3. **Net Displacement in Vertical Direction:**
   \[
   \text{Net Vertical} = 200 \, \text{m} - 500 \, \text{m} = -300 \, \text{m}
   \]

### Calculating the Magnitude of the Net Displacement Vector:
The magnitude of the net displacement vector, \(\vec{S}_{\text{net}}\), is given by:

\[
|\vec{S}_{\text{net}}| = \sqrt{(\text{Net East-West})^2 + (\text{Net North-South})^2 + (\text{Net Vertical})^2}
\]

Substituting the calculated values:

\[
|\vec{S}_{\text{net}}| = \sqrt{(1200)^2 + (5700)^2 + (-300)^2} \, \text{m}
\]

Let's calculate this.The magnitude of the net displacement for the day is approximately \( 5832.67 \, \text{m} \).

Would you like more details or have any further questions? Here are some related questions:

1. How do you decompose vectors into their components?
2. What is the difference between displacement and distance?
3. How do you add vectors graphically and analytically?
4. Why is the vertical component important in calculating the net displacement?
5. How would the net displacement change if one of the afternoon displacements were reversed?

**Tip:** When dealing with vectors, always break them down into their components and use the Pythagorean theorem to find the resultant magnitude.

What is the magnitude of your net displacement for the day?

Learn how to calculate the net displacement of a hiking trip using vector addition. This problem involves decomposing displacements into components and applying the Pythagorean theorem to find the magnitude of the net displacement. Suitable for students in Grades 9-12.

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