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WWW.HWX66.COM:高一数学教学课件平面向量数量积的坐标表示_图文

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5.5 线段的定比分点
直线l上两点 P1 、P2 ,在l上取不同于 P1 、 P2 的任一点P,则 P点与 P1 P2 的位置有哪几种情形? P在之间 P1 P2 ,P在 P1 P2 的延长线上, P在P2 P1的延长线上.

P1

P

P2

P1

P2

P

P

P1
?1 ? ? ? 0

P2

? ?0

? ? ?1

存在一个实数λ,使 P1 P ? ? PP2 ,λ叫做点P分有向线 段 P1 P2 所成的比.
能根据P点的三种不同的位置和实数与向量的积的向量 方向确定λ 的取值范围吗? www.niuwk.com 牛牛文库文档分享

5.5 线段的定比分点
设 P1 ( x1 , y1 ), P2 ( x2 , y2 ) ,P分 P1 P2 所成的比为 ? ,如何 求P点的坐标呢?
?    P1 P ? ( x ? x1 , y ? y1 ) PP2 ? ( x2 ? x , y2 ? y ) P1 P ? ? PP2

?    ( x ? x1 , y ? y1 ) ? ? ( x2 ? x, y2 ? y)
x1 ? ?x 2 ? x? ? ? x ? x1 ? ? ( x2 ? x ) ? 1? ? ?    ? ? ? y ? y ? ? ( y ? y ) ? 1 2 ? y ? y1 ? ?y2 ? ? 1? ? www.niuwk.com 牛牛文库文档分享

5.5 线段的定比分点
有向线段 P1 P2 的定比分点坐标公式
x1 ? ?x 2 ? x ? ? ? 1? ? ? ? y ? y1 ? ?y 2 ? ? 1? ?

有向线段 P1 P2 的中点坐标公式

x1 ? x 2 ? ?x ? 2 ? ? y ? y1 ? y2 牛牛文库文档分享 ?www.niuwk.com2

5.5 线段的定比分点
例题讲解
1 例1.已知两点 P1 (3,2) , P2 (?8,3) ,求点 P ( , y ) 分P1 P2 所 2 成的比 ? 及 y 的值.

解:由线段的定比分点坐标公式,得
? 1 3 ? ? ? ( ?8) ? ? ?2 1? ? ? ?y ? 2???3 ? ? 1? ?

解得

5 ? ?? ? 17 ? ?y ? 2 5 ? 22

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5.5 线段的定比分点
?ABC 的三个顶点的坐标分别为 A( x1 , y1 ) , 例2.如图, C ( x 3 , y 3 ),D是边AB的中点,G是CD上的一点, B( x2 , y2 ) , CG y ? 2,求点G的坐标. 且 D A GD
解:∵D是AB的中点 B x1 ? x 2 y1 ? y2 ( , ) ∴点D的坐标为 2 2 CG ?   ? 2 ?  CG ? 2GD GD 由定比分点坐标公式可得G点坐标为: G C O x

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5.5 线段的定比分点
解:∵D是AB的中点

∴点D的坐标为 (

CG B ?   ? 2 ?  CG ? 2GD GD 由定比分点坐标公式可得G点坐标为: x1 ? x 2 x3 ? 2 ? x1 ? x 2 ? x 3 2 x? ? 1? 2 3 y ? y2 y3 ? 2 ? 1 y1 ? y2 ? y3 2 y? ? 1? 2 3 x1 ? x2 ? x3 y1 ? y2 ? y3 ( , ) 即点G的坐标为www.niuwk.com 牛牛文库文档分享 3 3

x1 ? x 2 y1 ? y2 , ) 2 2

y D G C O

A

x

5.5 线段的定比分点
2 (1)如图,点B 分有向线段 AC 的比为?1 ? _____ 3 ,点C 2 ? 分有向线段BA 的比为 ?2 ? _____ 5 ,点A分有向线段BC 的比 3 ? ? ? _____ 为 3 5 . A B C

练习:

(2)连结A(4,1)和B(-2,4)两点的直线,和x轴 ) y轴交点的坐标是 (0, 3) 交点的坐标是 (6,0 ,和 .

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5.5 线段的定比分点
练习:
?ABC 中,AB的中点是D(-2,1),AC的中 (3)如图,
点是E(2,3),重心是G(0,1),求A、B、C的坐标. A D G B C E

A(0,5),B(-4,-3),C(4,1)

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