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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 a=xi+yj=(x,y). (3)向量的长度:即向量的大小,记作|a|. (4)特殊的向量:零向量a=O|a|=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(x1,y1)=(x2,y2) (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a∥b.平行向量也称为共线向量. 3.向量的运算
4.重要定理、公式 (1)平面向量基本定理 e1,e2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a=λ1e1+λ2e2. (2)两个向量平行的充要条件 a∥b[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]a=λb(b≠0)[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]x1y2-x2y1=O. (3)两个向量垂直的充要条件 a⊥b[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]x1x2+y1y2=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) 当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(x,y)按向量a=(h,k)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y=f(x)按向量a=(h,k)平移后所得的曲线的函数解析式为: y-k=f(x-h) (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:a2=b2+c2-2bccosA, b2=c2+a2-2cacosB, c2=a2+b2-2abcosC. (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. ①S△=1/2aha=1/2bhb=1/2chc ②S△=Pr ③S△=abc/4R ④S△=1/2sinC·ab=1/2ac·sinB=1/2cb·sinA ⑤S△=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image094.gif[/img] [海伦公式] ⑥S△=1/2(b+c-a)ra[如下图]=1/2(b+a-c)rc=1/2(a+c-b)rb [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I为S△ABC的内心, S△=Pr 图2中的I为S△ABC的一个旁心,S△=1/2(b+c-a)ra 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=a,AC=b,AB=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) ②BN=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image114.gif[/img]=1/2(a+c-b) ③FC=[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Rt△ABC,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,D是BC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AD是BC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AD是BC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] |
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沙发#
发布于:2019-02-24 08:34
希望对大家有帮助
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