**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 Hiker's Path**: - 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 now (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. Here, the eastward movement (3 km) and the northward movement (4 km) form the two sides of a right triangle. \[ \text{Displacement} = \sqrt{(3 \text{ km})^2 + (4 \text{ km})^2} \] \[ = \sqrt{9 + 16} \] \[ = \sqrt{25} \] \[ = 5 \text{ km} \] 4. **Direction of Displacement**: The direction of the displacement can also be calculated using trigonometry. The angle θ with respect to the eastward direction can be found using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \] To find θ, we can use the arctangent function: \[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \] This gives us the angle of the displacement vector relative to the eastward direction. ### Why Other Options Are Incorrect - **Option B (7 km)**: This option might confuse distance with displacement. The total distance traveled by the hiker is 3 km + 4 km = 7 km, but this is not the displacement, which is the straight-line distance from the start to the end point. - **Option C (1 km)**: This option is incorrect because it underestimates the straight-line distance between the starting and ending points. The displacement is not simply the difference in the east and north distances. - **Option D (12 km)**: This option is also incorrect as it does not relate to any calculation of distance or displacement in this scenario. It may arise from a misunderstanding of how to calculate the total distance traveled versus the displacement. ### Common Pitfalls - **Confusing Distance with Displacement**: Remember that distance is the total path length traveled, while displacement is the shortest path between two points. - **Forgetting to Use the Pythagorean Theorem**: When dealing with right triangles, always consider using the Pythagorean theorem to find the hypotenuse (displacement). - **Neglecting Direction**: While the question specifically asks for the magnitude of displacement, it's important to remember that displacement is a vector and has both magnitude and direction. ### 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 are at right angles. - The correct displacement for the hiker's journey is 5 km, calculated from the coordinates (0,0) to (3,4). - Always differentiate between total distance traveled and displacement to avoid confusion.