Find the magnitude and direction of each vector. Leave magnitude as a simplified radical. Round direction to the nearest tenth of a degree. \( \vec{g}=\langle 9,-7\rangle \)
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To find the magnitude of the vector \(\vec{g} = \langle 9, -7 \rangle\), we use the formula for the magnitude of a vector, which is \(\sqrt{x^2 + y^2}\). Calculating this, we get: \[ \text{Magnitude} = \sqrt{9^2 + (-7)^2} = \sqrt{81 + 49} = \sqrt{130} \] Thus, the magnitude is \(\sqrt{130}\). Now, for the direction (angle \(\theta\)), we use the tangent function: \[ \tan(\theta) = \frac{y}{x} = \frac{-7}{9} \] To find \(\theta\), we compute the arctangent: \[ \theta = \tan^{-1}\left(-\frac{7}{9}\right) \] Calculating this gives: \[ \theta \approx -38.7^\circ \] Since the vector lies in the fourth quadrant, we can find the positive angle by adding \(360^\circ\): \[ \theta \approx 360^\circ - 38.7^\circ \approx 321.3^\circ \] So, the magnitude of \(\vec{g}\) is \(\sqrt{130}\) and the direction is approximately \(321.3^\circ\). When it comes to vectors, historical figures like Galilei and Newton set the stage for understanding motion through vector analysis. While some explorers once risked their lives traversing uncharted territories without precise direction, today, we can navigate easily with vector calculus, showcasing the strides made in knowledge! In practical situations, combining vectors can lead to common mistakes, such as forgetting to consider the signs of the components. Always remember to revert back to your coordinate system: the direction you choose affects both angle and resultant vector significantly, especially in navigation or physics contexts!
