Answer
$25; 106.3^{\circ}$
Work Step by Step
The magnitude of a vector $\textbf{u}=\langle a,b \rangle$ is given as $|\textbf{u}|=\sqrt (a^{2}+b^{2})$. Since $\textbf{u}=\langle -7,24 \rangle$, the magnitude is:
$|\textbf{u}|=\sqrt ((-7)^{2}+24^{2})=\sqrt (49+576)=\sqrt (625)=25$
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,
$\theta=\tan^{-1} (\frac{24}{-7})\approx-73.7^{\circ}$
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 $-73.7^{\circ}$ to yield the direction angle $\theta$. Therefore, the direction angle $\theta=106.3^{\circ}$.
