**Correct Option: A. 5 km** ### Step-by-Step Explanation 1. **Understanding Displacement**: Displacement is a vector quantity that refers to the shortest distance from the initial position to the final position, along with the direction. It is different from distance, which is a scalar quantity that only considers the total path traveled. 2. **Visualizing the Problem**: - The hiker first walks 3 km east. We can represent this movement on a coordinate system where the starting point is at the origin (0,0). After walking east, the hiker's position is (3,0). - Next, the hiker walks 4 km north. From the position (3,0), moving north means adding 4 km to the y-coordinate. The new position is (3,4). 3. **Calculating Displacement**: - To find the displacement, we need to determine the straight-line distance from the starting point (0,0) to the final position (3,4). - We can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the two sides of the triangle are the distances traveled east (3 km) and north (4 km). \[ \text{Displacement} = \sqrt{(3 \text{ km})^2 + (4 \text{ km})^2} \] \[ = \sqrt{9 \text{ km}^2 + 16 \text{ km}^2} \] \[ = \sqrt{25 \text{ km}^2} \] \[ = 5 \text{ km} \] 4. **Direction of Displacement**: The direction of the displacement can also be calculated using trigonometry. The angle θ with respect to the east direction can be found using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4 \text{ km}}{3 \text{ km}} \] This gives us the angle θ, but for the purpose of this question, we are primarily interested in the magnitude of the displacement, which is 5 km. ### Why Other Options Are Incorrect - **Option B (7 km)**: This option might confuse distance traveled with displacement. The total distance the hiker walked is 3 km + 4 km = 7 km, but this is not the displacement, which is the straight-line distance. - **Option C (1 km)**: This option is incorrect as it significantly underestimates the straight-line distance between the starting and ending points. The hiker's path forms a right triangle, and the displacement is calculated using the Pythagorean theorem, which clearly shows a larger distance. - **Option D (12 km)**: This option is also incorrect as it does not relate to the actual path taken or the displacement. It may arise from a misunderstanding of how to calculate displacement versus total distance. ### Common Pitfalls - Confusing distance with displacement: Remember that distance is the total path length, while displacement is the shortest straight-line distance. - Misapplying the Pythagorean theorem: Ensure that you are using the correct sides of the triangle formed by the movements. - Forgetting to consider direction: While this question focuses on magnitude, in other contexts, direction is crucial for understanding displacement. ### Revision Summary - Displacement is the shortest distance from the starting point to the final position, represented as a vector. - Use the Pythagorean theorem to calculate displacement when movements form a right triangle. - The correct displacement for the hiker's journey is 5 km. - Always differentiate between total distance traveled and displacement to avoid confusion.