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 of the vector

The vector has components \(x = 5\) and \(y = 6\). Using the formula for the magnitude of a vector \(\vec{v}=(x,y)\), \(|\vec{v}|=\sqrt{x^{2}+y^{2}}\).

$$|\vec{v}|=\sqrt{5^{2}+6^{2}}=\sqrt{25 + 36}=\sqrt{61}$$

Step2: Calculate the direction angle \(\theta\)

Using the formula \(\tan\theta=\frac{y}{x}\), where \(x = 5\) and \(y = 6\). So \(\tan\theta=\frac{6}{5}=1.2\).
Then \(\theta=\arctan(1.2)\approx50^{\circ}\) (since \(\arctan(1)\approx45^{\circ}\) and \(1.2>1\))

Answer:

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