题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-96730.png)
(1)求证:MN⊥平面ABN;(2)求二面角A—BN—C的余弦值
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-91257.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-44143.png)
解析
试题分析:(Ⅰ)建立空间直角坐标系,求出向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061516-92495.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061516-11615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061516-93060.png)
(II)求平面NBC的法向量,平面ABN的法向量,利用向量的数量积求得二面角A-BN-C的余弦值.
解:法一 :以A点为原点,AB为x轴,AD为y轴,AD为z轴的空间直角坐标系,
则依题意可知相关各点的坐标分别是A(0,0,0),B(
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-96730.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-96730.png)
D(0,1,0),S(0,0,1)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061516-36186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061516-63385.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061517-15271.png)
∴MN⊥平面ABN.………………………………………6分
(2)设平面NBC的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061517-50705.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061517-61285.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061517-85770.png)
令a=1,则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061517-35970.png)
显然,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061518-12692.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061518-71728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061518-63799.png)
法二:(1)由题意知
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061518-22977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061519-30839.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061519-72184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061519-58552.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061519-48906.png)
(2)因为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061519-31792.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061520-10459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061520-12679.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061520-50897.png)
又在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061520-69077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061521-84851.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061521-64174.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061515-44143.png)
点评:解决该试题的关键是合理的建立空间直角坐标系,然后准确的表示点的坐标,和法向量的坐标,进而得到垂直的判定和二面角的平面角的求解。
核心考点
试题【(本题满分12分)如图所示,已知四棱锥S—ABCD的底面ABCD是矩形,M、N分别是CD、SC的中点,SA⊥底面ABCD,SA=AD=1,AB=.(1)求证:M】;主要考察你对线线角等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061509-18766.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061509-80272.png)
(Ⅰ) 求证:PA∥平面MDB;
(Ⅱ) N为AP的中点,求CN与平面MBD所成角的正切值.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061504-53767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061504-12555.png)
A.若m⊥α,n⊥β,α⊥β,则m⊥n | B.若m//α,n//β,α//β,则m//n |
C.若m⊥α,n//β,α⊥β,则m⊥n | D.若m//α,n⊥β,α⊥β,则m//n |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061449-62332.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061449-97019.png)
(1)求证:MN //平面PAD (2)求点B到平面AMN的距离
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061449-34437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061439-85264.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061439-51406.png)
A.当![]() ![]() ![]() |
B.当![]() ![]() ![]() |
C.当![]() ![]() ![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-36541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-57122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-61909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-90047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-97410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-36361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-24944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-50467.png)
(Ⅰ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-39723.png)
(Ⅱ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-77664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-82242.png)
(Ⅲ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-17253.png)
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