A mammalogist tracking Arctic fox movements records that a fox travels 4.2 km northeast, then 5.6 km due southeast. Approximating northeast as 45° from north and southeast as 45° from south, use vector addition to find the straight-line distance from the starting point to the final position.

["Tracking the Arctic Fox: Vector Analysis Reveals Its Final Position Relative to Starting Point", "When studying Arctic fox movements, researchers rely on precise tracking to understand survival strategies and migration patterns in extreme environments. A recent GPS-based study by a dedicated mammalogist documents how one Arctic fox traversed two distinct paths: first 4.2 km northeast, then 5.6 km due southeast. To interpret this motion accurately, vector addition helps calculate the fox’s final displacement from its starting point. This article explains how to model these movements using polar coordinates and vector components to find the straight-line distance—the bearing and magnitude from origin to final location.", "### Understanding Directional Components", "Arctic foxes traverse landscapes defined by cardinal and intercardinal directions, but for scientific precision, we define northeast and southeast using standard angular conventions.", "- Northeast (N45°E): This corresponds to 45° east of north.\n In standard mathematical terms (measured counterclockwise from the positive x-axis), this direction is:\n $$\n \ heta_1 = 90^\circ - 45^\circ = 45^\circ\n $$\n So the vector points along 45° from the north axis, or equivalently, 45° east of north.", "- Southeast (S45°E): This corresponds to 45° east of south, equivalent to 45° below the positive x-axis (or -45° in standard angle measurement):\n $$\n \ heta_2 = -45^\circ \quad \ ext{or} \quad 315^\circ\n $$", "We use these standard polar angles to break each leg of the journey into x (east-west) and y (north-south) components.", "### Breaking Down the First Leg: 4.2 km Northeast", "Let vector (\vec{v_1}) = 4.2 km at 45°.", "$$\nv_{1x} = 4.2 \cos(45^\circ) = 4.2 \ imes \frac{\sqrt{2}}{2} \approx 4.2 \ imes 0.7071 = 2.97 \ ext{ km (east)}\n$$\n$$\nv_{1y} = 4.2 \sin(45^\circ) = 4.2 \ imes \frac{\sqrt{2}}{2} \approx 2.97 \ ext{ km (north)}\n$$", "So,\n$$\n\vec{v_1} \approx (2.97, 2.97) \ ext{ km}\n$$", "### Breaking Down the Second Leg: 5.6 km Southeast", "Let vector (\vec{v_2}) = 5.6 km at –45°.", "$$\nv_{2x} = 5.6 \cos(-45^\circ) = 5.6 \ imes \frac{\sqrt{2}}{2} \approx 5.6 \ imes 0.7071 = 3.96 \ ext{ km (east)}\n$$\n$$\nv_{2y} = 5.6 \sin(-45^\circ) = -5.6 \ imes \frac{\sqrt{2}}{2} \approx -3.96 \ ext{ km (south)}\n$$", "So,\n$$\n\vec{v_2} \approx (3.96, -3.96) \ ext{ km}\n$$", "### Adding the Vectors", "Total displacement vector (\vec{r}) is:", "$$\n\vec{r} = \vec{v_1} + \vec{v_2} = (2.97 + 3.96, 2.97 - 3.96) = (6.93, -0.99) \ ext{ km}\n$$", "This means the fox ends up 6.93 km east and 0.99 km south of the starting point.", "### Calculating the Straight-Line Distance", "The straight-line distance (d) from the start to the final position is the magnitude of the resultant vector:", "$$\nd = \sqrt{(6.93)^2 + (-0.99)^2} = \sqrt{48.02 + 0.98} = \sqrt{49.00} = 7.00 \ ext{ km}\n$$", "### Conclusion", "Using vector addition, we found that the Arctic fox’s net displacement is approximately 7.00 km in a direction barely south of east—about 6.98° south of east. This precise calculation helps mammalogists model movement efficiency, energy expenditure, and habitat use in the Arctic tundra, highlighting how small directional differences can influence survival in extreme climates.", "For researchers tracking marine and terrestrial mammals in polar regions, combining GPS data with vector analysis remains essential for accurate ecological insights.", "---", "Keywords: Arctic fox, mammalogist, vector addition, northeast movement, southeast movement, GPS tracking, displacement, straight-line distance, polar coordinates, ecological movement analysis."]