Question: In a city grid, $\|\overrightarrow{OA}\| = 5$ km and $\|\overrightarrow{OB}\| = 12$ km, with an angle of $90^\circ$ between them. If $\overrightarrow{OC} = 2\overrightarrow{OA} - \overrightarrow{OB}$, find $\|\overrightarrow{OC}\|$.
["Finding the Magnitude of Vector $\overrightarrow{OC}$ in a City Grid", "When navigating city grids, understanding vector directions and magnitudes is crucial—especially when calculating distances and directions from key points. This problem explores vector operations in a city layout where points $A$ and $B$ define orthogonal axes, helping determine the direct distance $|\overrightarrow{OC}|$ based on given vector relationships.", "Given:\n- $|\overrightarrow{OA}| = 5$ km\n- $|\overrightarrow{OB}| = 12$ km\n- Angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $90^\circ$\n- $\overrightarrow{OC} = 2\overrightarrow{OA} - \overrightarrow{OB}$", "We seek $|\overrightarrow{OC}|$, the distance from the origin $O$ to point $C$ defined by this vector expression.", "---", "### Using the Pythagorean Principle for Perpendicular Vectors", "Since $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular (angle $90^\circ$), the magnitude of a linear combination like $\overrightarrow{OC} = 2\overrightarrow{OA} - \overrightarrow{OB}$ follows the Pythagorean theorem:", "$$\n|\overrightarrow{OC}| = \sqrt{(2|\overrightarrow{OA}|)^2 + (|\overrightarrow{OB}|)^2}\n$$", "Substitute the given lengths:", "$$\n|\overrightarrow{OC}| = \sqrt{(2 \ imes 5)^2 + (12)^2}\n= \sqrt{10^2 + 12^2}\n= \sqrt{100 + 144}\n= \sqrt{244}\n$$", "Simplify $\sqrt{244}$:", "$$\n\sqrt{244} = \sqrt{4 \ imes 61} = 2\sqrt{61}\n$$", "---", "### Final Result", "Thus, the magnitude of vector $\overrightarrow{OC}$ is:", "$$\n|\overrightarrow{OC}| = 2\sqrt{61} \ ext{ km}\n$$", "This precise measurement aids urban planners, GPS systems, and pedestrians in accurately estimating distances in grid-based city landscapes.", "---", "Keywords: vector magnitude, city grid vector math, $ |\overrightarrow{OC}| $, perpendicular vectors, $ 2\overrightarrow{OA} - \overrightarrow{OB} $, geometry in urban planning, coordinate-free vector calculation", "Meta Description:\nCalculate the distance $|\overrightarrow{OC}|$ where $\overrightarrow{OC} = 2\overrightarrow{OA} - \overrightarrow{OB}$, $|\overrightarrow{OA}| = 5$, $|\overrightarrow{OB}| = 12$, and angle $90^\circ$ between them. Learn how to compute vector magnitudes using the Pythagorean theorem in city grid navigation."]
