구 위 점 내적 최댓값

$\mathrm{\vec{AP}} \cdot \vec{AQ} = \left( \vec{AC} + \vec{CP} \right) \cdot \vec{AQ}$$=$$\mathrm{\vec{AC}} \cdot {\vec{AQ}} + {\vec{CP}} \cdot {\vec{AQ}}$ $\displaystyle \mathrm{\vec{CP}} \cdot \vec{AQ} \leq \sqrt{2} \times 2 = 2 \sqrt{2}$ $\mathrm{\vec{AO}}$와 $\mathrm{\vec{AQ}}$가 이루는 각의 크기는 $\displaystyle \frac{\pi}{6}$ $\mathrm{\vec{OA}}$와 $\mathrm{\vec{AC}}$가 이루는 각의 크기를 $\alpha$라 하면 $\mathrm{\vec{AC}} = ( 0 , - 3 , - 5 )$, $\mathrm{\vec{AO}} = ( 0 , - 3 , - 3 )$ $\displaystyle \cos \alpha = \frac{4}{\sqrt{17}}$, $\displaystyle \sin \alpha = \frac{1}{\sqrt{17}}$ $\displaystyle \cos \left( \frac{\pi}{6} - \alpha \right) = \frac{4 \sqrt{3} + 1}{2 \sqrt{17}}$ $\mathrm{\vec{AC}} \cdot \vec{AQ} \leq$$\displaystyle \sqrt{34} \times 2 \times \left( \frac{4 \sqrt{3} + 1}{2 \sqrt{17}} \right)$$\displaystyle = 4 \sqrt{6} + \sqrt{2}$ 따라서 최댓값은 $\displaystyle ( 4 \sqrt{6} + \sqrt{2} ) + 2 \sqrt{2} = 4 \sqrt{6} + 3 \sqrt{2}$, $\therefore$ $p + q = 3 + 4 = 7$