已知M,N,P三点的坐标分别为(-1,0),(3,-1),(1,2),并且向量ME=1/3向量MP,向量NF=1/3向量NP1,求点E,F的坐标2,证明向量EF‖向量MN

已知M,N,P三点的坐标分别为(-1,0),(3,-1),(1,2),并且向量ME=1/3向量MP,向量NF=1/3向量NP1,求点E,F的坐标2,证明向量EF‖向量MN
已知M,N,P三点的坐标分别为(-1,0),(3,-1),(1,2),并且向量ME=1/3向量MP,向量NF=1/3向量NP
1,求点E,F的坐标
2,证明向量EF‖向量MN

已知M,N,P三点的坐标分别为(-1,0),(3,-1),(1,2),并且向量ME=1/3向量MP,向量NF=1/3向量NP1,求点E,F的坐标2,证明向量EF‖向量MN
(1)
ME = 1/3 MP and NF = 1/3NP
Let O be the origin
OE = OM + ME
=(-1,0)+ ME
=(-1,0) + 1/3MP (ME=1/3MP)
=(-1,0) + 1/3(MO + OP)
=(-1,0) + 1/3(-OM + OP)
=(-1,0) + 1/3[ -(-1,0)+ (1,2) ]
=(-1/3,2/3)
i.e E(-1/3,2/3)
OF = ON + NF
= (3,-1) + 1/3(NP)
= (3,-1) + 1/3(-ON+OP)
= (3,-1) + 1/3[ -(3,-1)+(1,2) ]
= (3,-1) + (-2/3,1)
= (7/3,0)
i.e F(7/3,0)
(2)
EF = EO + OF
= -(-1/3,2/3) + (7/3,0)
= (8/3,-2/3)
MN = MO + ON
= -(-1,0) + (3,-1)
=(4,-1)
Ratio of x co-ordinate ,y co-ordinate of EF = (8/3)/(-2/3) = -4
Ratio of x co-ordinate ,y co-ordinate of MN = 4/-1 = -4
Therefore MN//EF

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