Answer
magnitude=$9.5208$
direction angle=$θ=119.0642^{\circ}$
Work Step by Step
The magnitude of a vector $\textbf{u}+\textbf{v}=⟨a,b⟩$ is given as $|\textbf{u}+\textbf{v}|=\sqrt (a^{2}+b^{2})$. Since $\textbf{u}+\textbf{v}=⟨−4.6251,8.3219⟩$, the magnitude is:
$|\textbf{u}+\textbf{v}|=\sqrt ((−4.6251)^{2}+(8.3219)^{2})=9.5208$
The direction angle $θ$ can be found through the equation $\tanθ=\frac{b}{a}$. Substituting the values of $a$ and $b$ in the formula and solving using a calculator,
$θ=\tan^{-1}(\frac{8.3219}{-4.6251})=−60.9358^{\circ}$
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 $−60.9358^{\circ}$ to yield the direction angle $θ$. Therefore, the direction angle $θ=119.0642^{\circ}$.
