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《利用向量法求空间角》教案.pdf

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1、 3.2.3 立体几何中的向量方法利用空间向量求空间角教学目标1.使学生学会求异面直线所成的角、直线与平面所成的角、二面角的向量方法；2.使学生能够应用向量方法解决一些简单的立体几何问题；3.使学生的分析与推理能力和空间想象能力得到提高.教学重点求解二面角的向量方法教学难点二面角的大小与两平面法向量夹角的大小的关系教学过程一、复习引入1用空间向量解决立体几何问题的“三步曲”（1）建立立体图形与空间向量的联系，用空间向量表示问题中涉及的点、直线、平面，把立体几何问题转化为向量问题；（化为向量问题）（2）通过向量运算，研究点、直线、平面之间的位置关系以及它们之间距离和夹角等问题；（进行向量运算）（

2、3）把向量的运算结果“翻译”成相应的几何意义。（回到图形）2向量的有关知识：（1）两向量数量积的定义：bababa,cos|（2）两向量夹角公式：|,cosbababa（3）平面的法向量：与平面垂直的向量a b O 二、知识讲解与典例分析知识点 1：面直线所成的角（范围：2,0(）（1）定义：过空间任意一点o 分别作异面直线a 与 b的平行线a 与 b，那么直线 a 与 b所成的锐角或直角，叫做异面直线a 与 b 所成的角.（2）用向量法求异面直线所成角设两异面直线a、b 的方向向量分别为a和b，问题 1：当a与b的夹角不大于90时，异面直线 a、b 所成的角与a和b的夹角的关系？问题2：a与

3、b的夹角大于90时，异面直线a、b 所成的角与a和b的夹角的关系？结论：异面直线a、b 所成的角的余弦值为|,cos|cosnmnmnm思考：在正方体1111DCBAABCD中，若1E与1F分别为11BA、11DC的四等分点，求异面直线1DF与1BE的夹角余弦值？（1）方法总结：几何法；向量法（2）11,cosBEDF与BEDF11,cos相等吗？（3）空间向量的夹角与异面直线的夹角有什么区别？例 1 如图，正三棱柱111CBAABC的底面边长为a,侧棱长为a2，求1AC和1CB所成的角.解法步骤：1.写出异面直线的方向向量的坐标。2.利用空间两个向量的夹角公式求出夹角。解：如图建立空间直角坐

4、标系xyzA，则)2,0(),0,21,23(),2,21,23(),0,0,0(11aaBaaCaaaCAObaObaba,ba,AxDCB1Azy1D1C1B1E1Fx y Z AyxCB1AD1B1C文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y

5、7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10

6、Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U1

7、0Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U

8、10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8

9、U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ

10、8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 H

11、J8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10)2,21,23(1aaaAC，)2,21,23(1aaaCB即21323|,cos22111111aaCBACCBACCBAC1AC和1CB所成的角为3练习 1：在 RtAOB 中，AOB=90，现将 AOB 沿着平面 AOB 的法向量方向平移到A1O1B1的位置，已知OA=OB=OO1，取 A1B1、A1O1的中点 D1、F1，求异面直线BD1与 AF1所成的角的余弦值。解：以点 O 为坐标原点建立空间直角坐标系，并设OA=1,则 A(1,0,0)，B(0,

12、1,0)，F1(21,0,1)，D1(21,21,1)1,0,21(1AF)1,21,21(,1BD103023451041|,cos111111BDAFBDAFBDAF所以，异面直线BD1 与 AF1 所成的角的余弦值为知识点 2、直线与平面所成的角（范围：2,0）思考：设平面的法向量为n，则BAn,与的关系？ABOnABOn（图 1）（图 2）文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E

13、5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2

14、E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K

15、2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4

16、K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H

17、4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9

18、H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ

19、9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10 x y Z AyxCB1AD1B1C据图分析可得：结论：例 2、如图，正三棱柱111CBAABC的底面边长为a,侧棱长为a2，求1AC和BBAA11面所成角的正弦值.分析：直线与平面所成的角步骤：1.求出平面的法向量2.求出直线的方向向量3.求以上两个向量的夹角，(锐角)其余角为所求角解：如图建立空间直角坐标系xyzA

20、，则),0,0(),2,0,0(1aABaAA)2,21,23(1aaaAC设平面BBAA11的法向量为),(zyxn由00002001zyayazABnAAn取1x，)0,0,1(n21323|,cos22111aaNACnACnAC1AC和BBAA11面所成角的正弦值21.练习：正方体1111DCBAABCD的棱长为1，点E、F分别为CD、1DD的中点.求直线11CB与平面CAB1所成的角的正弦值.BAn,22,BAn|,cos|sinABn131(,2)22ACaaa文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10

21、HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10

22、 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O1

23、0 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O

24、10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4

25、O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N

26、4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1

27、N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10知识点 3：二面角（范围：,0）方向向量法：将二面角转化为二面角的两个面的方向向量（在二面角的面内且垂直于二面角的棱）的夹角。如图，设二面角l的大小为，其中CDlCDABlAB,.结论：例 3

28、、如图，甲站在水库底面上的点A 处，乙站在水坝斜面上的点B 处.从 A，B 到直线（库底与水坝的交线）的距离 AC 和 BD 分别为a 和 b,CD 的长为 c,AB 的长为 d.求库底与水坝所成二面角的余弦值.解：如图.dABcCDbBDaAC，根据向量的加法法则,.DBCDACAB222)(DBCDACABd)(2222DBCDDBACCDACBDCDACDBACbca2222DBCAbca2222D C B A l|,coscosCDABCDABCDABBxADC1Bzy1A1D1CAB文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3

29、D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL

30、3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:C

31、L3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:

32、CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码

33、:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编

34、码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档

35、编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10于是，得22222dcbaDBCA设向量CA与DB的夹角为，就是库与水坝所成的二面角.因此.c o s22222dcbaab所以.2c o s2222abdcba库底与水坝所

36、成二面角的余弦值是.22222abdcba法向量法结论：或归纳：法向量的方向：一进一出，二面角等于法向量夹角；同进同出，二面角等于法向量夹角的补角.例 4、如图，ABCD是一直角梯形，90ABC，SA面ABCD，1BCABSA，1nl 2n21,nn21,coscosnn21,coscosnn1nl 2n21,nn21,nn21,nnABCzS文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R

37、10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5

38、R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E

39、5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2

40、E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K

41、2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4

42、K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H

43、4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R1021AD，求面SCD与面SBA所成二面角的余弦值.解：如图建立空间直角坐标系xyzA，则)1,0,0(),0,21,0(),0,1,1(),0,0,0(SDCA易知面SBA的法向量为)0,21,0(1ADn)1,21,0(),0,21,1(SDCD设面SCD的法向量为),(2zyxn，则有0202zyyx，取1z，得2,1

44、yx，)1,21,1(2n36|,cos212121nnnnnn又1n方向朝面内，2n方向朝面外，属于“一进一出”的情况，二面角等于法向量夹角即所求二面角的余弦值为36.练习：正方体1111DCBAABCD的棱长为1，点E、F分别为CD、1DD的中点.求二面角DAEF的余弦值。解：由题意知，)0,1,21(),21,1,0(EF，则)21,1,0(AF)0,1,21(,AE设平面AEF的法向量为),(zyxn，则02102100yxzyAEnAFn，取1y，得2zx)2,1,2(n又平面AED的法向量为)1,0,0(1AAABxDC1Bzy1A1D1CEF文档编码:CL3D4M1N4O10 H

45、J8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10

46、HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10

47、 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O1

48、0 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O

49、10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4

50、O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N4O10 HJ8U10Y7Q2S2 ZQ9H4K2E5R10文档编码:CL3D4M1N

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