Answer
The magnitude of $u+v$ is 9.5
The direction of $u+v$ is $119^{\circ}$ (measured counterclockwise from the positive x-axis).
Work Step by Step
Let $u = 12$
Let $v = 3$
Let angle $\theta$ be the angle between these two vectors. Then $\theta = 260^{\circ}-110^{\circ} = 150^{\circ}$
Let $c = u+v$. We can use $u$ and $v$ to complete the parallelogram. Note that the angle $C$ opposite the side $c$ has a value of $C = 180^{\circ}-150^{\circ} = 30^{\circ}$.
We can use the law of cosines to find $c$, the magnitude of $u+v$:
$c^2 = a^2+b^2-2ab~cos~C$
$c = \sqrt{a^2+b^2-2ab~cos~C}$
$c = \sqrt{(12)^2+(3)^2-(2)(12)(3)~cos~30^{\circ}}$
$c = \sqrt{90.646}$
$c = 9.5$
The magnitude of $u+v$ is 9.5
Let $V$ be the angle between $u$ and $u+v$. We can use the law of sines to find the angle $V$:
$\frac{c}{sin~C} = \frac{v}{sin~V}$
$sin~V = \frac{v~sin~C}{c}$
$sin~V = \frac{3~sin~30}{9.5}$
$sin~V = 0.15789$
$V = arcsin(0.15789)$
$V = 9^{\circ}$
The direction of $u+v$ is $110^{\circ}+9^{\circ}$ which is $119^{\circ}$ measured counterclockwise from the positive x-axis.
