평행사변형 벡터 최댓값

두 선분 ${\mathrm{AC}}$와 ${\mathrm{BD}}$의 교점을 ${\mathrm{O}}$라 하면 $\left| {\vec{\mathrm{PA}}} + {\vec{\mathrm{PB}}} + {\vec{\mathrm{PC}}} + {\vec{\mathrm{PD}}} \right|$ $= \left| {\vec{\mathrm{OA}}} + {\vec{\mathrm{OB}}} + {\vec{\mathrm{OC}}} + {\vec{\mathrm{OD}}} - 4 {\vec{\mathrm{OP}}} \right|$ $= \left| \left( {\vec{\mathrm{OA}}} + {\vec{\mathrm{OC}}} \right) + \left( {\vec{\mathrm{OB}}} + {\vec{\mathrm{OD}}} \right) - 4 {\vec{\mathrm{OP}}} \right|$ $= \left| - 4 {\vec{\mathrm{OP}}} \right| = 4 \left| {\vec{\mathrm{OP}}} \right|$ $\left| {\vec{\mathrm{BD}}} \right| ^{2}$$= \left( {\vec{\mathrm{CD}}} - {\vec{\mathrm{CB}}} \right) \bullet \left( {\vec{\mathrm{CD}}} - {\vec{\mathrm{CB}}} \right)$ $= \left| {\vec{\mathrm{CD}}} \right| ^{2} + \left| {\vec{\mathrm{CB}}} \right| ^{2}$$- 2 \left| {\vec{\mathrm{CD}}} \right| \left| {\vec{\mathrm{CB}}} \right| \cos \left( \angle {\mathrm{BCD}} \right)$ $\displaystyle = 6 ^{2} + 4 ^{2} - 2 \times 6 \times 4 \times \left( - \frac{1}{4} \right)$ $= 64$ $\left| {\vec{\mathrm{BD}}} \right| = 8$ $\displaystyle \left| {\vec{\mathrm{PA}}} + {\vec{\mathrm{PB}}} + {\vec{\mathrm{PC}}} + {\vec{\mathrm{PD}}} \right| = \frac{1}{2} \left| {\vec{\mathrm{BD}}} \right|$에서 $\displaystyle 4 \left| {\vec{\mathrm{OP}}} \right| = \frac{1}{2} \left| {\vec{\mathrm{BD}}} \right| = 4$이므로 $\left| {\vec{\mathrm{OP}}} \right| = 1$ 점 ${\mathrm{P}}$는 점 ${\mathrm{O}}$를 중심으로 하고 반지름의 길이가 $1$인 원 위의 점이다.

${\vec{\mathrm{AQ}}} = {\vec{\mathrm{AC}}} - {\vec{\mathrm{AP}}} = {\vec{\mathrm{PC}}}$이므로 ${\vec{\mathrm{PB}}} \bullet {\vec{\mathrm{DQ}}}$$= {\vec{\mathrm{PB}}} \bullet \left( {\vec{\mathrm{AQ}}} - {\vec{\mathrm{AD}}} \right)$ $= {\vec{\mathrm{PB}}} \bullet \left( {\vec{\mathrm{PC}}} - {\vec{\mathrm{AD}}} \right)$ $= {\vec{\mathrm{PB}}} \bullet \left( {\vec{\mathrm{PC}}} + {\vec{\mathrm{CB}}} \right)$ $= {\vec{\mathrm{PB}}} \bullet {\vec{\mathrm{PB}}}$ $= \left( {\vec{\mathrm{OB}}} - {\vec{\mathrm{OP}}} \right) \bullet \left( {\vec{\mathrm{OB}}} - {\vec{\mathrm{OP}}} \right)$ $= \left| {\vec{\mathrm{OB}}} \right| ^{2} - 2 \left( {\vec{\mathrm{OB}}} \bullet {\vec{\mathrm{OP}}} \right) + \left| {\vec{\mathrm{OP}}} \right| ^{2}$ $= 16 - 2 \left( {\vec{\mathrm{OB}}} \bullet {\vec{\mathrm{OP}}} \right) + 1$ $= 17 - 2 \left( {\vec{\mathrm{OB}}} \bullet {\vec{\mathrm{OP}}} \right)$ $\left| {\vec{\mathrm{BD}}} \right| = 8$이므로 $\left| {\vec{\mathrm{OB}}} \right| = 4$ 두 벡터 ${\vec{\mathrm{OB}}}$, ${\vec{\mathrm{OP}}}$가 이루는 각의 크기를 $\theta \left( 0 \leq \theta \leq \pi \right)$라 하면 ${\vec{\mathrm{PB}}} \bullet {\vec{\mathrm{DQ}}}$$= 17 - 2 \left| {\vec{\mathrm{OB}}} \right| \left| {\vec{\mathrm{OP}}} \right| \cos \theta$ $= 17 - 2 \times 4 \times 1 \times \cos \theta$ $= 17 - 8 \cos \theta$ 따라서 ${\vec{\mathrm{PB}}} \bullet {\vec{\mathrm{DQ}}}$의 최댓값은 $\theta = \pi$일 때, $17 - 8 \cos \pi = 25$ [참고] ${\vec{\mathrm{AQ}}} = {\vec{\mathrm{AC}}} - {\vec{\mathrm{AP}}}$ ${\vec{\mathrm{OQ}}} - {\vec{\mathrm{OA}}} = {\vec{\mathrm{OC}}} - {\vec{\mathrm{OA}}} - \left( {\vec{\mathrm{OP}}} - {\vec{\mathrm{OA}}} \right)$ ${\vec{\mathrm{OQ}}} = {\vec{\mathrm{OA}}} + {\vec{\mathrm{OC}}} - {\vec{\mathrm{OP}}} = - {\vec{\mathrm{OP}}}$