题目内容
在四棱锥P-ABCD中,底面ABCD是边长为1的正方形,PA
底面ABCD,点M是棱PC的中点,AM
PBD.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231431404101829.gif)
(1)求PA的长
(2)证明PB
平面AMD
(3)求棱PC与平面AMD所成角
的余弦值.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140363106.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140363106.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231431404101829.gif)
(1)求PA的长
(2)证明PB
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140363106.gif)
(3)求棱PC与平面AMD所成角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140441193.gif)
1,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140456254.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140456254.gif)
解:(1)首先建好空间直角坐标系,以A为原点,
AB为x轴,AD为y轴,AP为z轴.
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140472333.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140519563.gif)
所以PA的长为1;
(2)先证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140534673.gif)
从而得PB
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140363106.gif)
(3)平面AMD的法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140566396.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140581484.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140597861.gif)
所以棱PC与平面AMD所成角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140441193.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143140456254.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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