【答案】
分析:(1)以D为坐标原点,以DA,DB,DC为x轴y轴z轴建立空间直角坐标系,见图①,利用
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/0.png)
的夹角余弦值求直线DB
1与BC
1夹角的余弦值.
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/images1.png)
(2)如图②
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/images2.png)
直线DB是直线B
1B在平面ABCD上的射影则AC⊥DB,根据三垂线定理,有AC⊥B
1B.过点A在平面ABB
1A
1内作AM⊥B
1B于M,连接MC,MO,由△AMB≌△CMB,得CM⊥BB
1
∠AMC是二面角A-B
1B-C的一个平面角,在三角形AMC中求出此角即可
解答:解:(1)以D为坐标原点,以DA,DB,DC为x轴y轴z轴建立空间直角坐标系.如图①
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/images3.png)
则各点坐标D(0,0,0)B(2,2.0)B1(1,1,2)C1(0,1,2)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/1.png)
=(1,1,2),
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/2.png)
=(-2.-1,2)
设
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/3.png)
的夹角为θ,则cosθ=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/4.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/5.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/6.png)
直线DB
1与BC
1夹角的余弦值为
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/7.png)
.
(2)如图②
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/images11.png)
∵直线DB是直线B
1B在平面ABCD上的射影,AC⊥DB,
根据三垂线定理,有AC⊥B
1B.
过点A在平面ABB
1A
1内作AM⊥B
1B于M,连接MC,MO,
由△AMB≌△CMB,得CM⊥BB
1
所以,∠AMC是二面角A-B
1B-C的一个平面角.
根据勾股定理,有
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/8.png)
.
∵OM⊥B
1B,有
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/9.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/10.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/11.png)
,
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/12.png)
.
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101225557240640360/SYS201311012255572406403018_DA/13.png)
.
点评:本小题主要考查直线与直线的夹角、二面角及其平面角等有关知识,考查空间想象能力和思维能力,属于中档题