【答案】
分析:解法一:幾何法
(I)根據(jù)直棱柱的幾何特征,結(jié)合∠B
1A
1C
1=90°,可證得A
1C
1⊥平面A
1B
1BA,進(jìn)而AD⊥A
1C
1,由勾股定理可得A
1D⊥AD,最后由線面垂直的判定定理得到AD⊥平面A
1DC;
(Ⅱ)連結(jié)AC
1交A
1C于點(diǎn)E,取AD的中點(diǎn)F,連結(jié)EF,則EF∥C
1D,∠CEF或它的補(bǔ)角就是異面直線C
1D與直線A
1C所成的角,解△CEF可得答案.
解法二:向量法
(I)以A為原點(diǎn)建立坐標(biāo)系,求出
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/0.png)
,
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/1.png)
,
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/2.png)
的坐標(biāo)后,根據(jù)向量垂直的充要條件,及線面垂直的判定定理可得AD⊥平面A
1DC;
(Ⅱ)求出直線C
1D與直線A
1C的方向向量,代入向量夾角公式,可得答案.
解答:![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/images3.png)
解法一:幾何法
證明:(Ⅰ)∵AA
1⊥平面A
1B
1C
1,
∴AA
1⊥A
1C
1又A
1C
1⊥A
1B
1,
∴A
1C
1⊥平面A
1B
1BA
∴AD⊥A
1C
1∵AD=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/3.png)
,A
1D=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/4.png)
,AA
1=2,
由AD
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/5.png)
,
得A
1D⊥AD
∵A
1C
1∩A
1D=A
1∴AD⊥平面A
1DC
1…(7分)
解:(Ⅱ)連結(jié)AC
1交A
1C于點(diǎn)E,取AD的中點(diǎn)F,連結(jié)EF,則EF∥C
1D
∴∠CEF或它的補(bǔ)角就是異面直線C
1D與直線A
1C所成的角
由(Ⅰ)知,AD⊥A
1C
1,則AD⊥AC,
又AF=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/6.png)
AD=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/7.png)
在△CEF中,CE=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/8.png)
,EF=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/9.png)
,CF=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/10.png)
cos∠CEF=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/11.png)
則異面直線C
1D與直線A
1C所成角的余弦值為
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/12.png)
…(14分)
解法二:以A為原點(diǎn)建立坐標(biāo)系,如圖,則A
1(0,0,2),C(0,1,0),C
1(0,1,2)
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/images14.png)
D(1,0,1)…(3分)
(Ⅰ)∵
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/13.png)
=( 1,0,-1 ),
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/14.png)
=( 1,0,1 ),
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/15.png)
=( 0,1,0 ),
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/16.png)
•
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/17.png)
=1+0-1=0,
∴A
1D⊥AD …(5分)
又
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/18.png)
•
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/19.png)
=0,∴AD⊥A
1C
1∵A
1D∩A
1C
1=A
1∴AD⊥A
1DC
1…(8分)
(Ⅱ)
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/20.png)
=(1,-1,-1),
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/21.png)
=(0,1,-2)
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/22.png)
=1
cos<
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/23.png)
>=
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/24.png)
故直線C
1D與直線A
1C所成角的余弦值
![](http://thumb.zyjl.cn//pic6/res/gzsx/web/STSource/20131103174519031861870/SYS201311031745190318618016_DA/25.png)
…(14分)
點(diǎn)評(píng):本題考查的知識(shí)點(diǎn)是直線與平面垂直的判定,異面直線及其所成的角,解法一的關(guān)鍵是(1)熟練掌握線線垂直,線面垂直,面面垂直之間的相互轉(zhuǎn)化,(2)將異面直線夾角轉(zhuǎn)化為解三角形問題,解法二的關(guān)鍵是建立空間坐標(biāo)系,將問題轉(zhuǎn)化為向量夾角問題.