Solution: Use the Pythagorean theorem since $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular. Compute $\|\overrightarrow{OC}\|^2 = (2\|\overrightarrow{OA}\|)^2 + (\|\overrightarrow{OB}\|)^2 = 4(25) + 144 = 100 + 144 = 244$. Thus, $\|\overrightarrow{OC}\| = \sqrt{244} = 2\sqrt{61}$. Final answer: $\boxed{2\sqrt{61}}$.
["Mastering Vector Calculations: How the Pythagorean Theorem Simplifies $|\overrightarrow{OC}|$", "When working with vectors in a coordinate plane, the Pythagorean theorem remains one of the most powerful and elegant tools—especially when dealing with perpendicular vectors. This article explores a practical problem where using the Pythagorean theorem significantly streamlines the calculation of vector magnitude, delivering a clear and efficient solution.", "Suppose we have two vectors, $\overrightarrow{OA}$ and $\overrightarrow{OB}$, which are perpendicular (i.e., $\angle AOB = 90^\circ$) and represented geometrically as $ |\overrightarrow{OA}| = 5 $ and $ |\overrightarrow{OB}| = 12 $. We seek to compute the magnitude squared of $ \overrightarrow{OC} = 2\overrightarrow{OA} + \overrightarrow{OB} $, a common vector combination in physics and engineering.", "Because $ \overrightarrow{OA} \perp \overrightarrow{OB} $, the vectors form a right triangle, allowing us to apply the Pythagorean theorem directly:", "$$\n|\overrightarrow{OC}|^2 = |2\overrightarrow{OA}|^2 + |\overrightarrow{OB}|^2\n$$", "First, scale the magnitude:", "$$\n|2\overrightarrow{OA}|^2 = (2 \cdot |\overrightarrow{OA}|)^2 = (2 \cdot 5)^2 = 10^2 = 100\n$$", "Then compute the perpendicular component:", "$$\n|\overrightarrow{OC}|^2 = 100 + 144 = 244\n$$", "Taking the square root to find the magnitude:", "$$\n|\overrightarrow{OC}| = \sqrt{244}\n$$", "But we can simplify $ \sqrt{244} $: since $ 244 = 4 \ imes 61 $,", "$$\n\sqrt{244} = \sqrt{4 \cdot 61} = 2\sqrt{61}\n$$", "Thus, the length of vector $ \overrightarrow{OC} $ is:", "$$\n\boxed{2\sqrt{61}}\n$$", "Why This Approach Works:\nUsing the Pythagorean theorem eliminates the need for complex coordinate systems or dot product expansions when vectors are orthogonal. This method not only saves time but also reduces computational error—making it ideal for problems in geometry, physics, and navigation.", "Whether you’re analyzing forces in mechanics or plotting trajectories, recognizing perpendicular vectors lets you apply Pythagoras directly—turning vectors into simplified, solvable equations.", "---", "Key Takeaways:\n- When two vectors are perpendicular, $|\overrightarrow{OC}|^2 = (k|\vec{OA}|)^2 + |\vec{OB}|^2$\n- Scaling a vector’s magnitude first simplifies the computation\n- Simplifying square roots like $ \sqrt{4 \cdot 61} $ yields clean radical forms", "Leverage the Pythagorean theorem to unlock faster, clearer vector solutions!"]
