题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045756-41011.jpg)
(1)求证:AD⊥BC;
(2)求二面角B—AC—D的余弦值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045756-85375.png)
解析
试题分析:(1)证明 作AH⊥平面BCD于H,连接BH、CH、DH,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045756-45694.jpg)
易知四边形BHCD是正方形,且AH=1,以D为原
点,以DB所在直线为x轴,DC所在直线为y轴,
以垂直于DB,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045756-20514.png)
标系,如图所示,则B(2,0,0),C(0,2,0),
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-80966.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-87343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-91664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-27462.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-70647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045758-28683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045758-84614.png)
因此
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-87343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045758-60876.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045758-45768.png)
(2)解:设平面ABC的法向量为n1=(x,y,z),则由n1⊥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-87343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045757-87343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045758-68650.png)
同理由n1⊥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045759-72775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045759-72775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045759-15075.png)
可取n1=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045759-40715.png)
同理,可求得平面ACD的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045759-76713.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045800-12974.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045800-11223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045800-43636.png)
即二面角B—AC—D的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045756-85375.png)
点评:本题考查线面垂直,考查面面角,解题的关键是掌握线面垂直的判定方法,正确运用向量法解决面面角问题.
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045751-99774.png)
如图,三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045731-39737.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045731-52747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045731-50148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045731-49792.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045731-85302.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045732-46902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045732-95966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045732-90319.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045732-97796.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045733-35269.png)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045732-97796.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045733-82489.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045725-89977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045725-23819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045725-66195.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045725-50276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045726-61445.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045726-93854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045726-61754.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-38289.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-73767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-79502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-73767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-95590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-51649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045717-81150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-46495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-83304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-37018.png)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-67152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-90228.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045718-27084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045719-84036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045719-21898.png)
A.若l![]() | B.若l⊥β且α∥β,则l⊥α |
C.若l⊥β且α⊥β,则l∥α | D.若α![]() |
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