试题分析:(Ⅰ)∵四边形
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521817510.png)
是菱形,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522129664.png)
.
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522144513.png)
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521895522.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522176433.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522191669.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522207622.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522222550.png)
.
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522238565.png)
, ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521973519.png)
.…………………2分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521832393.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521817510.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522285428.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521817510.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521832393.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522004395.png)
.又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522020579.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522035415.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
,………………………………………4分
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522285428.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522425461.png)
,
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521910475.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
. ………………………………6分
(Ⅱ)解法一:由(1)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522035415.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522285428.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522534448.png)
,
∴平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522534474.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
………………………7分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521832393.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521817510.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522597542.png)
.
由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522222550.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522628603.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522644416.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522534448.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522675427.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
,
∴平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522737478.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522534448.png)
.…………………………9分
∴平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522534448.png)
是平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
的公垂面.
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522878501.png)
就是平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
所成的锐二面角的平面角.……10分
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522924604.png)
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522940835.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522956490.png)
.……………11分
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522987453.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240035230021021.png)
.
所以,平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
所成的锐二面角的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522113461.png)
.…………14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240035230655503.png)
理(Ⅱ)解法二:以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523080298.png)
为原点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523096378.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522004395.png)
分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523143262.png)
轴、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523158311.png)
轴的正方向,建立空间直角坐标系
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523174517.png)
,如图所示.因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521864560.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521895522.png)
,所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523221520.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523236551.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523236631.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523252619.png)
,…………7分
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523268743.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523299643.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523299745.png)
.………8分
由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522035415.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
,
故平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
的一个法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523361591.png)
.……………………9分
设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
的一个法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523392657.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240035234081013.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523424911.png)
,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523439415.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003523455712.png)
. …………………11分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240035234701503.png)
.
所以,平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521926439.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003521957456.png)
所成的锐二面角的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003522113461.png)
.……14分
点评:典型题,立体几何题,是高考必考内容,往往涉及垂直关系、平行关系、角、距离、体积的计算。在计算问题中,有“几何法”和“向量法”。利用几何法,要遵循“一作、二证、三计算”的步骤,本题解法较多二应用向量则简化了证明过程。