Find the displacement of a soccer player who runs 15m [E 15° N] then 13m [N 5° W].

To find the displacement of the soccer player, we need to break down the given vectors into their horizontal (east-west) and vertical (north-south) components and then combine them to find the resultant displacement.

### Step 1: Break down each vector into components

1. **First vector: 15 m [E 15° N]**
   - This means 15 meters at 15° north of east.
   - Horizontal (east) component:
     \[
     x_1 = 15 \times \cos(15°) = 15 \times 0.9659 = 14.49 \text{ m (East)}
     \]
   - Vertical (north) component:
     \[
     y_1 = 15 \times \sin(15°) = 15 \times 0.2588 = 3.88 \text{ m (North)}
     \]

2. **Second vector: 13 m [N 5° W]**
   - This means 13 meters at 5° west of north.
   - Horizontal (west) component:
     \[
     x_2 = 13 \times \sin(5°) = 13 \times 0.0872 = 1.13 \text{ m (West)}
     \]
   - Vertical (north) component:
     \[
     y_2 = 13 \times \cos(5°) = 13 \times 0.9962 = 12.95 \text{ m (North)}
     \]

### Step 2: Combine the components

- **Total horizontal component** (East-West):
  \[
  x_{\text{total}} = 14.49 \text{ m (East)} - 1.13 \text{ m (West)} = 13.36 \text{ m (East)}
  \]

- **Total vertical component** (North-South):
  \[
  y_{\text{total}} = 3.88 \text{ m (North)} + 12.95 \text{ m (North)} = 16.83 \text{ m (North)}
  \]

### Step 3: Find the resultant displacement

Now, use the Pythagorean theorem to find the magnitude of the resultant displacement:
\[
\text{Displacement} = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2}
\]
\[
\text{Displacement} = \sqrt{(13.36)^2 + (16.83)^2} = \sqrt{178.49 + 283.34} = \sqrt{461.83} = 21.49 \text{ m}
\]

### Step 4: Find the angle of displacement

The angle of displacement (relative to the east direction) is given by:
\[
\theta = \tan^{-1} \left(\frac{y_{\text{total}}}{x_{\text{total}}}\right) = \tan^{-1} \left(\frac{16.83}{13.36}\right) = \tan^{-1} (1.26) = 51.7°
\]

### Final Answer:
The displacement of the soccer player is **21.49 m [E 51.7° N]**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is vector addition, and how does it apply to displacement problems?
2. How do you calculate the angle of a vector relative to a specific direction?
3. Can displacement ever be larger than the total distance traveled?
4. How do the sine and cosine functions relate to vector components?
5. What is the significance of breaking a vector into components?

**Tip**: Always visualize vectors in a coordinate system to help with understanding their direction and components.

Discover how to calculate the displacement of a soccer player using vector addition and trigonometry. This problem involves breaking down vectors into components, applying the Pythagorean theorem, and calculating the resultant displacement with angles. Suitable for Grades 9-11.

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