Beginner Vector Geometry #3

Note that $\overrightarrow{a}\cdot\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos\theta$ .

From the question, we see that $\overrightarrow{a}\cdot\overrightarrow{b}=45-32=13$ .

Also, $|\overrightarrow{a}|=\sqrt{3^2+(-4)^2}=5$ and $|\overrightarrow{b}|=\sqrt{15^2+8^2}=17$ .

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Therefore,

$\begin{aligned} \cos\theta&=\dfrac{|\overrightarrow{a}||\overrightarrow{b}|\cos\theta}{|\overrightarrow{a}||\overrightarrow{b}|} \\ &=\dfrac{\overrightarrow{a}\cdot\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|} \\ &=\boxed{\dfrac{13}{85}} \end{aligned}$

$\therefore~p+q=85+13=\boxed{98}$ .