Question: In a city map, vectors $ \overrightarrow{OA} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} $ and $ \overrightarrow{OB} = \begin{bmatrix} -1 \\ 3 \end{bmatrix} $ represent street segments. Find $ \overrightarrow{OC} $ such that $ \overrightarrow{OC} = 2\overrightarrow{OA} - 3\overrightarrow{OB} $.
["Title: How to Calculate Vector OC from Vectors OA and OB in a City Map", "In urban planning, vector analysis plays a crucial role in representing street directions and connections on a city map. Vectors help define paths, distances, and spatial relationships visually and mathematically. Imagine vectors $ \overrightarrow{OA} = \begin{bmatrix} 2 \ 1 \end{bmatrix} $ and $ \overrightarrow{OB} = \begin{bmatrix} -1 \ 3 \end{bmatrix} $ as key street segments originating from intersection point $ O $. A common question arises in such mapping: Find vector $ \overrightarrow{OC} $ defined as $ \overrightarrow{OC} = 2\overrightarrow{OA} - 3\overrightarrow{OB} $. Let’s break this down step by step.", "### Understanding the Vector Operation", "The expression $ \overrightarrow{OC} = 2\overrightarrow{OA} - 3\overrightarrow{OB} $ means we scale vector $ \overrightarrow{OA} $ by 2 and subtract three times vector $ \overrightarrow{OB} $. This operation models real-world applications—such as comparing two street directions with different magnitudes and directions in a grid layout.", "### Step-by-Step Calculation", "1. Scale vector $ \overrightarrow{OA} $ by 2:\n$$\n2\overrightarrow{OA} = 2 \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix}\n$$", "2. Scale vector $ \overrightarrow{OB} $ by 3:\n$$\n3\overrightarrow{OB} = 3 \begin{bmatrix} -1 \ 3 \end{bmatrix} = \begin{bmatrix} -3 \ 9 \end{bmatrix}\n$$", "3. Subtract the scaled vectors:\n$$\n\overrightarrow{OC} = \begin{bmatrix} 4 \ 2 \end{bmatrix} - \begin{bmatrix} -3 \ 9 \end{bmatrix} = \begin{bmatrix} 4 + 3 \ 2 - 9 \end{bmatrix} = \begin{bmatrix} 7 \ -7 \end{bmatrix}\n$$", "### Final Result", "The resulting vector is\n$$\n\overrightarrow{OC} = \begin{bmatrix} 7 \ -7 \end{bmatrix}\n$$", "This vector represents a new direction and magnitude on the city map, derived through linear combination of the original street segments $ \overrightarrow{OA} $ and $ \overrightarrow{OB} $. Such vector computations assist urban planners and navigators in analyzing intersecting road segments efficiently.", "---", "By applying basic vector algebra as shown here, city maps gain clarity and precision—turning abstract paths into clear, actionable spatial data. Whether designing efficient routes or teaching geographic concepts, understanding vector operations like $ \overrightarrow{OC} = 2\overrightarrow{OA} - 3\overrightarrow{OB} $ enhances how we interpret and build urban spaces.", "Keywords: vector OC, urban planning, city map, vector calculation, $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, linear algebra applied to geography, street segment vectors, coordinate geometry in cities, vector subtraction, vector scaling."]
