Answer
$|\textbf{v}|=10$
$\theta=126.9^{\circ}$
Work Step by Step
The magnitude of a vector $\textbf{v}=\langle a,b \rangle$ is given as $|\textbf{v}|=\sqrt (a^{2}+b^{2})$. Since $\textbf{v}=\langle -6,8 \rangle$, the magnitude is:
$|\textbf{v}|=\sqrt ((-6)^{2}+(8)^{2})=\sqrt (36+64)=\sqrt (100)=10$
The direction angle $\theta$ can be found through the equation $\tan\theta=\frac{b}{a}$. Substituting the values of $a$ and $b$ in the formula and solving using a calculator,
$\theta=\tan^{-1} (\frac{8}{-6})=\tan^{} (\frac{4}{-3})=-53.1^{\circ}$
The vector has a negative horizontal component and a positive vertical component which places it in the second quadrant. Since the direction angle is supposed to be the positive angle between the x-axis and the position vector, we need to add $180^{\circ}$ to $-53.1^{\circ}$ to yield the direction angle $\theta$. Therefore, the direction angle $\theta=126.9^{\circ}$.
