设=(x,y),。∵点P在直线OM上,∴与共线,而=(2,1), ∴=,即x=2y,则有=(2y,y),∵=-=(1,7)-(2y,y)=(1-2y,7-y),=-=(5,1)-(2y,y)=(5-2y,1-y), ∴·=(1-2y,7-y)·(5-2y,1-y)=(1-2y)(5-2y)+(7-y)(1-y) =5y2-20y+12=5(y-2)2-8,从而,当且仅当y=2,x=4时,·取得最小值-8,此时=(4,2),=(-3,5),=(1,-1), 于是此时||=,||=,·=-8, ∴cosÐAPB===-. |