What is the magnitude of $\langle 3,2\rangle+\langle 1,3\rangle$? Round your answer to the nearest tenth.
To find the magnitude of the sum of two vectors \(v_1 = \langle 3, 2 \rangle\) and \(v_2 = \langle 1, 3 \rangle\), we first perform vector addition. The sum of two vectors is found by adding their corresponding components:
\[v = v_1 + v_2 = \langle 3 + 1, 2 + 3 \rangle = \langle 4, 5 \rangle.\]
The magnitude of a vector \(v = \langle x, y \rangle\) is given by the formula \(||v|| = \sqrt{x^2 + y^2}\). Substituting the components of our resultant vector \(v = \langle 4, 5 \rangle\) into this formula, we get:
\[||v|| = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}.\]
We calculate the numerical value of \(\sqrt{41}\). Using the calculation, we find that \(\sqrt{41} \approx 6.403\). Rounding this value to the nearest tenth, we obtain \(6.4\). Comparing this result with the given options, we identify option b as the correct one. Therefore, the magnitude is \(\boxed{6.4}\).
Vector addition and the calculation of magnitude are fundamental in fields like physics and engineering, particularly when determining the resultant force of multiple objects pushing or pulling on a single point. For example, if a ship is pushed by ocean currents in one direction and wind in another, finding the magnitude of the resulting vector helps navigate the ship and determine its true speed. Similarly, it is used in computer graphics to calculate the length of paths or the distance an object travels in 2D space.