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
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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
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](/uploads/image/z/3819214-46-4.jpg?t=%E5%B7%B2%E7%9F%A5M%2CN%2CP%E4%B8%89%E7%82%B9%E7%9A%84%E5%9D%90%E6%A0%87%E5%88%86%E5%88%AB%E4%B8%BA%28-1%2C0%29%2C%283%2C-1%29%2C%281%2C2%29%2C%E5%B9%B6%E4%B8%94%E5%90%91%E9%87%8FME%3D1%2F3%E5%90%91%E9%87%8FMP%2C%E5%90%91%E9%87%8FNF%3D1%2F3%E5%90%91%E9%87%8F)
(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
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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