分析:先把六个面分为三组,在一组组的进行研究,找到直线与截面法向量的夹角即可得到结论.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082318230346711181.png)
解:首先,把六个面分成三组,AA
1D
1D和BB
1C
1C对截面的关系是一样的,其他四个是一样的,
以点D为原点,AD所在直线为X轴,DC所在直线为Y轴,DD
1所在直线为Z轴,
设正方体棱长为2;
则A(2,0,0),D(0,0,0),B(2,2,0),C(0,2,0),E(2,1,0),
F(0,1,2),A
1(2,0,2),B
1(2,2,2,),C
1(0,2,2),D
1(0,0,2);
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303482427.png)
=(-2,1,0),
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303498467.png)
=((0,1,2),
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303513421.png)
=(-2,2,0),
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303654399.png)
=(-2,-2,0),
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303670452.png)
=(-2,0,-2),
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303685447.png)
=(0,-2,-2);
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303701460.png)
=(0,2,-2)
因为要想面对角线截面A
1ECF成60°角,需要直线与法向量的夹角为30度,即其余弦值为±
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
.
设截面A
1ECF的法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
=(x,y,z),
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231823037481076.png)
?
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303763872.png)
?
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
=(1,2,1),且|
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304216341.png)
,
因为cos<
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303513421.png)
>=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304262785.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304278751.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304309464.png)
≠±
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
;
cos<
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303654399.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
>=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304278751.png)
=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
,
cos<
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303670452.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
>=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304278751.png)
≠±
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
;
cos(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303685447.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
>=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304278751.png)
=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
;
cos<
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303701460.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303732315.png)
>=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182304278751.png)
≠±
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823182303716453.png)
;
再看AA
1D
1D这个面里,
AD
1与EF平行,不是,
所以,一共四条.
故选:C.