Chapter 6, Problem 51RE

To determine

To find : The magnitude and the direction angle of the vector $v$ .

Explanation of Solution

Given: The vector is $v=7\left(120°i+\sin 120°j\right)$ .

The magnitude of the vector is calculated as,

  $\begin{array}{c}\left|\left|v\right|\right|=\sqrt{{\left(7\cos 120°\right)}^2+{\left(7\sin 120°\right)}^2}\\ =\sqrt{12.25+36.75}\\ =\sqrt{49}\\ =7\end{array}$

The direction of the vector is calculated as,

  $\begin{array}{c}\tan \theta =\frac{\sin 120°}{\cos 120°}\\ =-1.732\\ \arctan \left(\tan \theta \right)=\arctan \left(-1.732\right)\\ \theta =60°\end{array}$

The direction of the angle vector is,

  $\begin{array}{c}\theta =180°-60°\\ =120°\end{array}$

Therefore, the magnitude of vector is $7$ and the direction angle of the vector is $120°$ .