A drone flies 15 kilometers northeast, then 20 kilometers southeast. Using vector components, what is the total horizontal displacement from the starting point in the east direction?
["Title: How to Calculate Horizontal Displacement Using Vector Components: A Drone’s 15 km Northeast and 20 km Southeast Flight", "When a drone travels 15 kilometers northeast followed by 20 kilometers southeast, understanding its total horizontal displacement in the east direction requires a clear grasp of vector components. This article breaks down the drone’s journey using vector math to reveal its precise eastward movement from the starting point.", "---", "### Understanding Northeast and Southeast Directions via Vectors", "The compass directions northeast (NE) and southeast (SE) correspond to specific angles on a coordinate plane:", "- Northeast (NE) is 45° east of north, or equivalently 45° from the positive y-axis toward the positive x-axis.\n- Southeast (SE) is 45° south of east, or 315° from true north (or –45° from the positive x-axis).", "We represent each flight segment as a vector using east (x) and north (y) components:", "- Northeast flight (15 km):\n - The angle with respect to the positive x-axis is 45°.\n - ( x_1 = 15 \cos(45°) = 15 \cdot \frac{\sqrt{2}}{2} \approx 10.61 \ ext{ km} ) (east)\n - ( y_1 = 15 \sin(45°) = 15 \cdot \frac{\sqrt{2}}{2} \approx 10.61 \ ext{ km} ) (north)", "- Southeast flight (20 km):\n - The angle with respect to the positive x-axis is –45° (-45° from positive x-axis or 315°).\n - ( x_2 = 20 \cos(-45°) = 20 \cdot \frac{\sqrt{2}}{2} = 10\sqrt{2} \approx 14.14 \ ext{ km} ) (east)\n - ( y_2 = 20 \sin(-45°) = 20 \cdot \left(-\frac{\sqrt{2}}{2}\right) \approx –14.14 \ ext{ km} ) (north) — movement south, so negative", "---", "### Calculating Total Eastward Displacement", "Total eastward (horizontal) displacement is:\n[\nx_{\ ext{total}} = x_1 + x_2 = 15 \cdot \frac{\sqrt{2}}{2} + 20 \cdot \frac{\sqrt{2}}{2} = (15 + 20) \cdot \frac{\sqrt{2}}{2} = 35 \cdot \frac{\sqrt{2}}{2} \approx 24.75 \ ext{ km}\n]", "So, the drone’s net displacement east from the starting point is approximately 24.75 kilometers.", "---", "### Why Vector Components Matter", "Relying on direct Euclidean distance or simple addition would misrepresent the true extent. These calculations isolate the eastwest component, ignoring vertical movement, and reveal the net horizontal path—critical for navigation, landing site planning, and energy optimization.", "---", "### Summary", "By resolving vector components:\n- 15 km NE → +10.61 km east\n- 20 km SE → +14.14 km east", "Total horizontal displacement east:\n[\n\boxed{35 \cdot \frac{\sqrt{2}}{2} \approx 24.75 \ ext{ km}}\n]", "Use vector analysis to uncover accurate movement patterns—essential for drone operations, robotics, and geographic information systems (GIS)."]
