Question

review the graph. which statement accurately describes the vector shown? magnitude of \\( \sqrt { 11 } \\) and direction angle equal to approximately \\( 40 ^ { \circ } \\) magnitude of \\( \sqrt { 11 } \\) and direction angle equal to approximately \\( 50 ^ { \circ } \\) magnitude of \\( \sqrt { 61 } \\) and direction angle equal to approximately \\( 40 ^ { \circ } \\) magnitude of \\( \sqrt { 61 } \\) and direction angle equal to approximately \\( 50 ^ { \circ } \\)

Explanation:

Step1: Calculate the magnitude

If the vector has components \((x = 6,y = 5)\) (assuming from the graph, as starting from origin \((0,0)\) to \((6,5)\)), the magnitude \(r\) of a vector \(\vec{v}=(x,y)\) is given by the formula \(r=\sqrt{x^{2}+y^{2}}\).
Substitute \(x = 6\) and \(y = 5\) into the formula: \(r=\sqrt{6^{2}+5^{2}}=\sqrt{36 + 25}=\sqrt{61}\).

Step2: Calculate the direction angle

The direction angle \(\theta\) of a vector \(\vec{v}=(x,y)\) is given by \(\tan\theta=\frac{y}{x}\).
Here, \(x = 6\) and \(y = 5\), so \(\tan\theta=\frac{5}{6}\).
Using a calculator, \(\theta=\arctan(\frac{5}{6})\approx39.8^{\circ}\approx40^{\circ}\) (if we consider the wrong - component assumption) or if we assume the vector from \((0,0)\) to \((5,6)\) (maybe mis - reading the graph coordinates in a wrong initial analysis, but if we recalculate with \(x = 5\) and \(y = 6\)):
Magnitude \(r=\sqrt{5^{2}+6^{2}}=\sqrt{25 + 36}=\sqrt{61}\), and \(\tan\theta=\frac{6}{5}\), \(\theta=\arctan(\frac{6}{5})\approx50.2^{\circ}\approx50^{\circ}\)

Answer:

magnitude of \(\sqrt{61}\) and direction angle equal to approximately \(50^{\circ}\)