A sail boat sails `2km` due east, `5km 37^(@)` south of east, and finally an unknown displacement. If the final displacement of the boat from the starting point is `6km` due east. Determine the third displacement.

Explanation

To solve the problem, we will break down the displacements of the sailboat into vector components and then find the unknown displacement.

### Step-by-Step Solution:

1. **Identify the Displacements**:
- The first displacement \( R_1 \) is \( 2 \, \text{km} \) due east.
- The second displacement \( R_2 \) is \( 5 \, \text{km} \) at an angle of \( 37^\circ \) south of east.
- The third displacement \( R_3 \) is unknown.
- The final displacement \( R_T \) is \( 6 \, \text{km} \) due east.

2. **Convert Displacements to Vector Form**:
- For \( R_1 \):
\[ R_1 = 2 \, \text{km} \, \hat{i} \]
- For \( R_2 \):
- The eastward (i) component:
\[ R_{2x} = 5 \cos(37^\circ) \]
- The southward (j) component (negative because it's south):
\[ R_{2y} = -5 \sin(37^\circ) \]
- Therefore,
\[ R_2 = 5 \cos(37^\circ) \hat{i} - 5 \sin(37^\circ) \hat{j} \]

3. **Calculate the Components of \( R_2 \)**:
- Using the approximate values \( \cos(37^\circ) \approx 0.798 \) and \( \sin(37^\circ) \approx 0.601 \):
\[ R_{2x} \approx 5 \times 0.798 \approx 3.99 \, \text{km} \]
\[ R_{2y} \approx -5 \times 0.601 \approx -3.005 \, \text{km} \]
- Thus,
\[ R_2 \approx 3.99 \hat{i} - 3.005 \hat{j} \]

4. **Set Up the Equation for Total Displacement**:
- The total displacement \( R_T \) can be expressed as:
\[ R_T = R_1 + R_2 + R_3 \]
- Substituting the known values:
\[ 6 \hat{i} = (2 \hat{i} + 3.99 \hat{i} - 3.005 \hat{j} + R_3) \]

5. **Combine the Components**:
- Combine the i components:
\[ 6 \hat{i} = (2 + 3.99) \hat{i} - 3.005 \hat{j} + R_3 \]
- This simplifies to:
\[ 6 \hat{i} = 5.99 \hat{i} - 3.005 \hat{j} + R_3 \]

6. **Solve for \( R_3 \)**:
- Rearranging gives:
\[ R_3 = (6 - 5.99) \hat{i} + 3.005 \hat{j} \]
- Thus,
\[ R_3 = 0.01 \hat{i} + 3.005 \hat{j} \]

7. **Final Result**:
- The third displacement \( R_3 \) is approximately:
\[ R_3 \approx 0.01 \, \text{km} \, \hat{i} + 3.005 \, \text{km} \, \hat{j} \]
- In vector form, this means the boat moves slightly east and predominantly south.