Question

20£®ÒÑÖª$\overrightarrow{OP}$=£¨2£¬1£©£¬$\overrightarrow{OA}$=£¨1£¬7£©£¬$\overrightarrow{OB}$=£¨5£¬1£©£®ÉèMÊÇÖ±ÏßOPÉϵÄÒ»µã£¨ÆäÖÐOΪ×ø±êÔ­µã£©£¬µ±$\overrightarrow{MA}$•$\overrightarrow{MB}$È¡×îСֵʱ£º
£¨1£©Çó$\overrightarrow{OM}$£»      
£¨2£©Éè¡ÏAMB=¦È£¬Çócos¦ÈµÄÖµ£®

Answer 1

·ÖÎö £¨1£©Éè$\overrightarrow{OM}$=t$\overrightarrow{OP}$£¬ÀûÓÃÏòÁ¿µÄ×ø±ê±íʾÇó³ö$\overrightarrow{OM}$¡¢$\overrightarrow{MA}$ºÍ$\overrightarrow{MB}$£¬¼ÆËã$\overrightarrow{MA}•\overrightarrow{MB}$È¡µÃ×îСֵʱtµÄÖµ¼´¿É£»
£¨2£©ÓÉƽÃæÏòÁ¿µÄÊýÁ¿»ýÇó¼Ð½ÇµÄÓàÏÒÖµ¼´¿É£®

½â´ð ½â£º£¨1£©¸ù¾ÝÌâÒ⣬Éè$\overrightarrow{OM}$=t$\overrightarrow{OP}$£¬t¡ÊR£¬---£¨2·Ö£©
Ôò$\overrightarrow{OM}$=£¨2t£¬t£©£¬
$\overrightarrow{MA}$=£¨1-2t£¬7-t£©£¬
$\overrightarrow{MB}$=£¨5-2t£¬1-t£©£»
¡à$\overrightarrow{MA}•\overrightarrow{MB}$=5t²-20t+12=5£¨t-2£©²-8£¬---£¨4·Ö£©
¡àµ±t=2ʱ£¬$\overrightarrow{MA}•\overrightarrow{MB}$×îС£¬Õâʱ$\overrightarrow{OM}$=£¨4£¬2£©£»---£¨6·Ö£©
£¨2£©ÓÉ$\overrightarrow{MA}$=£¨-3£¬5£©£¬$\overrightarrow{MB}$=£¨1£¬-1£©£¬---£¨8·Ö£©
¡àcos¦È=$\frac{\overrightarrow{MA}•\overrightarrow{MB}}{|\overrightarrow{MA}|¡Á|\overrightarrow{MB}|}$=$\frac{-3¡Á1+5¡Á£¨-1£©}{\sqrt{{£¨-3£©}^{2}{+5}^{2}}¡Á\sqrt{{1}^{2}{+£¨-1£©}^{2}}}$=-$\frac{4\sqrt{17}}{17}$£¬
¡àcos¦ÈµÄÖµÊÇ$-\frac{{4\sqrt{17}}}{17}$£®---£¨12·Ö£©

µãÆÀ ±¾Ì⿼²éÁËƽÃæÏòÁ¿µÄÊýÁ¿»ýÓëÓ¦ÓÃÎÊÌ⣬Ҳ¿¼²éÁËÏòÁ¿¹²ÏßµÄÓ¦ÓÃÎÊÌ⣬ÊÇ»ù´¡ÌâÄ¿£®