已知O,A,B是平面内不共线的三点,满足向量OP=A*向量OA+B*向量OB,则P,A,B三点共线的充要条件是A+B=?

题目

已知O,A,B是平面内不共线的三点,满足向量OP=A*向量OA+B*向量OB,则P,A,B三点共线的充要条件是A+B=?

答案

P,A,B三点共线,则存在唯一实数t,使得向量PA=tPB,
(OA-OP)=t(OB-OP),
(t-1)OP=-OA+tOB,
OP=-1/(t-1)OA+t/(t-1)OB,
则a=-1/(t-1),b= t/(t-1)
a+b=-1/(t-1)+ t/(t-1)=1.
则P,A,B三点共线的充要条件是a+b=1.