题目内容
(12分) 已知在正方体ABCD —A1B1C1D1中,E、F分别是D1D、BD的中点,G在棱CD上,且CG =
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231632188534430.jpg)
(1)求证:EF⊥B1C;
(2)求EF与G C1所成角的余弦值;
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218838293.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231632188534430.jpg)
(1)求证:EF⊥B1C;
(2)求EF与G C1所成角的余弦值;
(1)略
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218869307.gif)
解:如图建立空间直角坐标系O—xyz,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231632188856897.jpg)
设正方体的棱长为4,则E (0,0,2),F (2,2,0),
C (0,4,0),B (4,4,0),C1(0,4,4),B1(4,4,4),G (0,3,0) . (2分)
(1)
,
∴
.
∴
. ∴EF⊥B1C. (5分)
(2)
,
∴
.
又∵
,
∴
. (10分)
因为,EF与GC1所成角的范围为(0,
]
所以,EF与GC1所成角的余弦值为
12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231632188856897.jpg)
设正方体的棱长为4,则E (0,0,2),F (2,2,0),
C (0,4,0),B (4,4,0),C1(0,4,4),B1(4,4,4),G (0,3,0) . (2分)
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218900710.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218931818.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218947444.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218963523.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218978779.gif)
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218994714.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231632190091171.gif)
因为,EF与GC1所成角的范围为(0,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163219025233.gif)
所以,EF与GC1所成角的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163218869307.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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