题目内容
(本小题满分12分)
棱长为1的正方体
中,P为DD1中点,O1、O2、O3分别为面
、面
、面
的中心。(1)求证:
。
(2)求异面直线PO3与O1O2所成角的余弦值。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824630510.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824661355.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824708332.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824739303.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824770366.gif)
(2)求异面直线PO3与O1O2所成角的余弦值。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231448248331333.gif)
解:(1)以D为坐标原点,DA、DC、DD1所在直线分别为
轴、
轴、
轴建坐标系,则
(1,1,1),
,P(0,0,
),A(1,0,0),
,
,
,
……………………………………(6分)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231448254571182.gif)
又
,故
……………(12分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824864187.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824895193.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144824911129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825082220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825145465.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825207225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825285540.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825332565.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825394757.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825394377.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231448254571182.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823144825488547.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231448248331333.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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