벡터 수직 코사인

그림과 같이 $\mathrm{\vec{BA}} = \mathit{\vec{a}}$, $\mathrm{\vec{BC}} = \mathit{\vec{c}}$라 놓으면 $\mathrm{\vec{BD}} =$$\displaystyle \frac{2}{3} \vec{a} + \frac{1}{3} \vec{c}$, $\mathrm{\vec{BE}} =$$\displaystyle \frac{1}{3} \vec{a} + \frac{2}{3} \vec{c}$, $\mathrm{\vec{BF}} =$$\displaystyle \frac{1}{2} \vec{c}$, $\mathrm{\vec{BG}}$는 실수 $t$, $s$에 대하여 $\displaystyle \frac{t}{3} \vec{a} + \frac{2 t}{3} \vec{c}$$\displaystyle = s \left( \frac{2}{3} \vec{a} + \frac{1}{3} \vec{c} \right) + ( 1 - s ) \left( \frac{1}{2} \vec{c} \right)$ $\displaystyle \frac{t}{3} = \frac{2 s}{3}$, $\displaystyle \frac{2 t}{3} = \frac{s}{3} + \frac{1 - s}{2}$, $\displaystyle t = \frac{2}{3}$, $\displaystyle s = \frac{1}{3}$, $\mathrm{\vec{BG}} =$$\displaystyle \frac{2}{9} \vec{a} + \frac{4}{9} \vec{c}$, $\mathrm{\vec{AG}} = \vec{BG} - \vec{BA} =$$\displaystyle - \frac{7}{9} \vec{a} + \frac{4}{9} \vec{c}$ $\mathrm{\vec{AG}} \cdot \vec{BE} =$$\displaystyle \left( - \frac{7}{9} \vec{a} + \frac{4}{9} \vec{c} \right) \cdot \left( \frac{1}{3} \vec{a} + \frac{2}{3} \vec{c} \right)$ $\displaystyle = \frac{1}{27} ( - 7 \vec{a} + 4 \vec{c} ) ( \vec{a} + 2 \vec{c} )$ $\displaystyle = \frac{1}{27} \left\{ - 7 | \vec{a} | ^{2} + 8 | \vec{c} | ^{2} - 10 ( \vec{a} \cdot \vec{c} ) \right\}$ $\displaystyle = \frac{1}{27} \{ - 63 + 128 - 10 ( \vec{a} \cdot \vec{c} ) \}$$= 0$ 따라서 $\displaystyle \vec{a} \cdot \vec{c} = \frac{13}{2}$, $\displaystyle \cos \theta = \frac{\vec{a} \cdot \vec{c}}{| \vec{a} | | \vec{c} |}$$\displaystyle = \frac{\frac{13}{2}}{3 \times 4} = \frac{13}{24}$$\displaystyle = \frac{q}{p}$ $\therefore$ $p + q = 37$