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[用户互动]高中数学所有知识点

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更多 发布于:2019-02-24 08:34
高中数学第五章-平面向量
考试内容:
 ©
向量.向量的加法与减法.实数与向量的积.平面向量的坐标表示.线段的定比分点.平面向量的数量积.平面两点间的距离、平移.
 ©考试要求:
 ©(1)理解向量的概念,掌握向量的几何表示,了解共线向量的概念.
 ©(2)掌握向量的加法和减法.
 ©(3)掌握实数与向量的积,理解两个向量共线的充要条件.
 ©(4)了解平面向量的基本定理,理解平面向量的坐标的概念,掌握平面向量的坐标运算.
 ©(5)掌握平面向量的数量积及其几何意义,了解用平面向量的数量积可以处理有关长度、角度和垂直的问题,掌握向量垂直的条件.
 ©(6)掌握平面两点间的距离公式,以及线段的定比分点和中点坐标公式,并且能熟练运用掌握平移公式.

§05. 平面向量  知识要点
1.本章知识网络结构

2.向量的概念
(1)向量的基本要素:大小和方向.(2)向量的表示:几何表示法 ;字母表示:a
坐标表示法 aj=().
(3)向量的长度:即向量的大小,记作|a|.
(4)特殊的向量:零向量aOa|=O.
单位向量aO为单位向量aO|=1.
(5)相等的向量:大小相等,方向相同(11)=(22
(6) 相反向量:a=-bb=-aa+b=0
(7)平行向量(共线向量):方向相同或相反的向量,称为平行向量.记作ab.平行向量也称为共线向量.
3.向量的运算
运算类型
几何方法
坐标方法
运算性质
向量的
加法

1.平行四边形法则
2.三角形法则




向量的
减法

三角形法则



,

1.是一个向量,满足:
2.>0时, 同向;
<0时, 异向;
=0时, .






是一个数
1.时,
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif[/img].
2.[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif[/img]

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.gif[/img]

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif[/img]

4.重要定理、公式
(1)平面向量基本定理
e1e2是同一平面内两个不共线的向量,那么,对于这个平面内任一向量,有且仅有一对实数λ1
λ2,使aλ1e1λ2e2.
(2)两个向量平行的充要条件
ab[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]aλb(b≠0)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]x1y2x2y1=O.
(3)两个向量垂直的充要条件
ab[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]a·b=O[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]x1x2y1y­2=O.
(4)线段的定比分点公式
设点P分有向线段[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image068.gif[/img]所成的比为λ,即[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image070.gif[/img]=λ[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image072.gif[/img],则
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.gif[/img]=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.gif[/img]+[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image076.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image080.gif[/img] (线段的定比分点的向量公式)
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image082.gif[/img] (线段定比分点的坐标公式)
λ=1时,得中点公式:
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.gif[/img]=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image084.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image078.gif[/img]+[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image080.gif[/img])或[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image086.gif[/img]
 (5)平移公式
设点P(xy)按向量a=()平移后得到点P′(x′,y′),
则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image088.gif[/img]=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image074.gif[/img]+a或[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image090.gif[/img]
曲线yfx)按向量a=()平移后所得的曲线的函数解析式为:
yfx)
(6)正、余弦定理
正弦定理:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image092.gif[/img]
余弦定理:a2b2c2-2bccosA
b2c2a2-2cacosB
c2a2b2-2abcosC.

 
(7)三角形面积计算公式:
设△ABC的三边为a,b,c,其高分别为ha,hb,hc半周长为P,外接圆、内切圆的半径为R,r.
S=1/2aha=1/2bhb=1/2chc                 S=Pr      S=abc/4R
S=1/2sinC·ab=1/2ac·sinB=1/2cb·sinA  S=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image094.gif[/img]
[海伦公式]  

S=1/2(b+c-ara[如下图]=1/2b+a-crc=1/2a+c-brb
[注]:到三角形三边的距离相等的点有4个,一个是内心,其余3个是旁心.
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image096.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image098.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image100.gif[/img]如图:                                          





[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image102.gif[/img]          
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image104.gif[/img]                [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image106.gif[/img]                       [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image108.gif[/img]


图1中的ISABC的内心, S=Pr
                                               
图2中的ISABC的一个旁心,S=1/2b+c-ara
                                                   
附:三角形的五个“心”;
重心:三角形三条中线交点.
外心:三角形三边垂直平分线相交于一点.
内心:三角形三内角的平分线相交于一点.
垂心:三角形三边上的高相交于一点.
旁心:三角形一内角的平分线与另两条内角的外角平分线相交一点.
⑸已知⊙O是△ABC的内切圆,若BC=aAC=bAB=c [注:s为△ABC的半周长,即[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image110.gif[/img]]
则:①AE=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image112.gif[/img]=1/2(b+c-a)                                              

BN=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image114.gif[/img]=1/2(a+c-b
FC=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image116.gif[/img]=1/2(a+b-c
综合上述:由已知得,一个角的邻边的切线长,等于半周长减去对边(如图4).                                  
特例:已知在RtABC,c为斜边,则内切圆半径r=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image118.gif[/img](如图3).            
⑹在△ABC中,有下列等式成立[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image120.gif[/img].
证明:因为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image122.gif[/img]所以[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image124.gif[/img],所以[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image126.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image128.gif[/img]结论!
⑺在△ABC中,DBC上任意一点,则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img].
证明:在△ABCD中,由余弦定理,有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image132.gif[/img]①
在△ABC中,由余弦定理有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image134.gif[/img]②,②代入①,化简
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image136.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image138.gif[/img]可得,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image130.gif[/img](斯德瓦定理)
①若ADBC上的中线,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image140.gif[/img];
②若AD是∠A的平分线,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image142.gif[/img],其中[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image144.gif[/img]为半周长;
③若ADBC上的高,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image146.gif[/img],其中[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image144.gif[/img]为半周长.
⑻△ABC的判定:
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image148.gif[/img]ABC为直角△[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]∠A + ∠B =[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image151.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image153.gif[/img]<[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image155.gif[/img]△ABC为钝角△[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]∠A + ∠B<[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image157.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image153.gif[/img]>[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image155.gif[/img]△ABC为锐角△[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]∠A + ∠B>[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image157.gif[/img]
附:证明:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image161.gif[/img],得在钝角△ABC中,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image163.gif[/img]
⑼平行四边形对角线定理:对角线的平方和等于四边的平方和.
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image165.gif[/img]

空间向量
1.空间向量的概念:
具有大小和方向的量叫做向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
注:⑴空间的一个平移就是一个向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
⑵向量一般用有向线段表示[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]同向等长的有向线段表示同一或相等的向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
⑶空间的两个向量可用同一平面内的两条有向线段来表示[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
2.空间向量的运算
定义:与平面向量运算一样,空间向量的加法、减法与数乘向量运算如下
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image169.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image171.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image173.gif[/img]
运算律:⑴加法交换律:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image175.gif[/img]
⑵加法结合律:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image177.gif[/img]
⑶数乘分配律:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image179.gif[/img]
3 [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]共线向量
表示空间向量的有向线段所在的直线互相平行或重合,则这些向量叫做共线向量或平行向量.[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]平行于[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]记作[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image185.gif[/img].
当我们说向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]、[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]共线(或[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]//[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img])时,表示[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]、[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]的有向线段所在的直线可能是同一直线,也可能是平行直线.
4.共线向量定理及其推论:
共线向量定理:空间任意两个向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]、[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]≠[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image188.gif[/img]),[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]//[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]的充要条件是存在实数λ,使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]=λ[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img].
推论:如果[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]为经过已知点A且平行于已知非零向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]的直线,那么对于任意一点O,点P在直线[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]上的充要条件是存在实数t满足等式
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image192.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]

其中向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]叫做直线[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]的方向向量.
5.向量与平面平行:
已知平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img]和向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img],作[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image198.gif[/img],如果直线[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image200.gif[/img]平行于[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img]或在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img]内,那么我们说向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]平行于平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img],记作:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image206.gif[/img].
通常我们把平行于同一平面的向量,叫做共面向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
说明:空间任意的两向量都是共面的[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
6.共面向量定理:
如果两个向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image208.gif[/img]不共线,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image210.gif[/img]与向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image208.gif[/img]共面的充要条件是存在实数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image213.gif[/img]使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image215.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
推论:空间一点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image217.gif[/img]位于平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image219.gif[/img]内的充分必要条件是存在有序实数对[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image213.gif[/img],使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image222.gif[/img]或对空间任一点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image224.gif[/img],有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image226.gif[/img]       ①
①式叫做平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image219.gif[/img]的向量表达式[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]


7 [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]空间向量基本定理:
如果三个向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image229.gif[/img]不共面,那么对空间任一向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image210.gif[/img],存在一个唯一的有序实数组[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.gif[/img],使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image234.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
推论:设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.gif[/img]是不共面的四点,则对空间任一点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image217.gif[/img],都存在唯一的三个
有序实数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.gif[/img],使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image239.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]
8 [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]空间向量的夹角及其表示:
已知两非零向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image208.gif[/img],在空间任取一点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image224.gif[/img],作[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image243.gif[/img],则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image245.gif[/img]叫做向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]的夹角,记作[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image249.gif[/img];且规定[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image251.gif[/img],显然有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image253.gif[/img];若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image255.gif[/img],则称[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]互相垂直,记作:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image258.gif[/img].
9.向量的模:
设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image198.gif[/img],则有向线段[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image261.gif[/img]的长度叫做向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]的长度或模,记作:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image264.gif[/img].
10.向量的数量积: [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image266.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image268.gif[/img].
已知向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image270.gif[/img]和轴[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image273.gif[/img]是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]上与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]同方向的单位向量,作点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image276.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]上的射影[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image278.gif[/img],作点[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image280.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]上的射影[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image282.gif[/img],则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image284.gif[/img]叫做向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image286.gif[/img]在轴[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image190.gif[/img]上或在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image273.gif[/img]上的正射影.
可以证明[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image284.gif[/img]的长度[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image288.gif[/img].
11.空间向量数量积的性质:    
(1)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image290.gif[/img].(2)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image292.gif[/img].(3)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image294.gif[/img].
12.空间向量数量积运算律:
(1)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image296.gif[/img].(2)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image298.gif[/img](交换律)(3)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image300.gif[/img](分配律).
空间向量的坐标运算

一.知识回顾:
(1)空间向量的坐标:空间直角坐标系的x轴是横轴(对应为横坐标),y轴是纵轴(对应为纵轴),z轴是竖轴(对应为竖坐标).
①令[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.gif[/img]=(a1,a2,a3),[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image304.gif[/img],则
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image306.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image308.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image310.gif[/img]    [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.gif[/img]∥[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image312.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image314.gif[/img]     [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.gif[/img]    
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image318.gif[/img](用到常用的向量模与向量之间的转化:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image320.gif[/img])
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image322.gif[/img]
②空间两点的距离公式:[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image324.gif[/img].
(2)法向量:若向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.gif[/img]所在直线垂直于平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image326.gif[/img],则称这个向量垂直于平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image326.gif[/img],记作[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image329.gif[/img],如果[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image329.gif[/img]那么向量[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image302.gif[/img]叫做平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image326.gif[/img]的法向量.
(3)用向量的常用方法:
①利用法向量求点到面的距离定理:如图,设n是平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image333.gif[/img]的法向量,AB是平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image333.gif[/img]的一条射线,其中[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image336.gif[/img],则点B到平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image333.gif[/img]的距离为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image338.gif[/img].
②利用法向量求二面角的平面角定理:设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image340.gif[/img]分别是二面角[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image342.gif[/img]中平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image344.gif[/img]的法向量,则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image340.gif[/img]所成的角就是所求二面角的平面角或其补角大小([img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image340.gif[/img]方向相同,则为补角,[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image340.gif[/img]反方,则为其夹角).
③证直线和平面平行定理:已知直线[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image346.gif[/img]平面[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image333.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image349.gif[/img],且CDE三点不共线,则a∥[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image333.gif[/img]的充要条件是存在有序实数对[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image352.gif[/img]使[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image354.gif[/img].(常设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image354.gif[/img]求解[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image357.gif[/img]若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image357.gif[/img]存在即证毕,若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image357.gif[/img]不存在,则直线AB与平面相交).
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image360.gif[/img]

一个愿意陪你到老的人!
lyh1807
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发布于:2019-02-24 08:34
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