Help with vectors and that stuff.

tan (theta) =  (7 / -8)  

arctan (7/-8) = theta  = -41.1859°

The terminal point of this vector lies in the second quadramt....so the angle is

180 - 41.1859  ≈ 138.81407° ≈  138. 8141°

And this is the angle ( theta ) that the vector makes with the positive x axis

1. The magnitude of  $\vec{u}$.

Round your answer to at least four decimal places.

$\vec{u} = \binom{-8}{7}$

$\begin{array}{|rcll|} \hline ||\vec{v}|| &=& \sqrt{(-8)^2+7^2} \\ &=& \sqrt{64+49} \\ &=& \sqrt{113} \\ &=& 10.6301458127 \\ &\approx& 10.6301 \\ \hline \end{array}$

2.The direction of $\vec{u}$, thaat is the angle $\theta$ it makes with the positive x-axis.

State your answer in degrees, rounded to at least four decimal places.

$\vec{u} = \binom{-8}{7}$

$\begin{array}{|rcll|} \hline \tan(\theta) &=& \frac{|~\vec{e_x} \times \vec{u}~| } {\vec{e_x} \cdot \vec{v} } \\ &=& \frac{ \left|~\binom{1}{0} \times \binom{-8}{7}~\right| } {\binom{1}{0} \cdot \binom{-8}{7} } \\ &=& \frac{ (1)\cdot (7) - (0)\cdot (-8) } { (1)\cdot (-8) + (0)\cdot (7) } \\ &=& \frac{ 7+0 } { -8+0 } \\ &=& \frac{ 7} { -8 } \quad & | \quad II.\text{Quadrant} \\ \theta &=& \arctan(\frac{ 7 } { -8 }) \\ \theta &=& \arctan(-0.875) \\ \theta &=& -41.1859251657^{\circ} + 180^{\circ} \quad & | \quad II.\text{Quadrant} \\ \theta &=& 138.814074834^{\circ} \\ \theta &\approx& 138.8141^{\circ} \\ \hline \end{array}$