(1)以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405845200.gif)
點(diǎn)為原點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405845235.gif)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540586085.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406001187.gif)
軸,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406016236.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540603272.gif)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406048193.gif)
軸,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406016236.gif)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406094129.gif)
軸的空間直角坐標(biāo)系,如圖所示.則依題意可知相關(guān)各點(diǎn)的坐標(biāo)分別是:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406110361.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406126408.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406141495.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406157361.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406250466.gif)
如下圖所示.………………………………………………………
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540626685.gif)
……………………(2分)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231854063915683.jpg)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406406379.gif)
點(diǎn)的坐標(biāo)分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406453905.gif)
……………………………
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540646972.gif)
……………(3分)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406484656.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406516568.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406531681.gif)
......................... (4分)
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231854065781009.gif" style="vertical-align:middle;" />,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406625483.gif)
.......................... (6分)
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231854066401230.gif" style="vertical-align:middle;" />,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406656494.gif)
.............. (7分)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405767393.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405782383.gif)
........................................................... (8分)
(2)設(shè)平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407015380.gif)
的法向量
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407030475.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407046552.gif)
,........................ (9分)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407249796.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407342710.gif)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407436701.gif)
............................................................. (10分)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407483577.gif)
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407514231.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540753071.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407576517.gif)
顯然,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185406484656.gif)
就是平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405782383.gif)
的法向量................................... (11分)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231854076862339.gif)
.................... (12分)
由圖形知,二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405798433.gif)
是鈍角二面角........................................ (13分)
所以二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407717432.gif)
的余弦值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407732284.gif)
.......................................... (14分)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231854077648743.jpg)
解:(1)取
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407779238.gif)
的中點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407795213.gif)
,連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407810436.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407904446.gif)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407935328.gif)
,所以四點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407951456.gif)
共面.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407966344.gif" style="vertical-align:middle;" />,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408044321.gif)
.......... (2分)]
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408060324.gif)
.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408076472.gif" style="vertical-align:middle;" />,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408091254.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408138269.gif)
..................... (4分)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408154317.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408169260.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408185415.gif)
................... (6分)
易證
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408216468.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405767393.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405782383.gif)
................... (8分)
(2)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408434234.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408450620.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408466553.gif)
.............................................................. (9分)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318540848172.gif)
同(1)可證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408497264.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408528270.gif)
.
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408606557.gif)
,且平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408622293.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408528270.gif)
.
明顯
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408637567.gif)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408684442.gif)
........................................... (10分)
過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185405845200.gif)
作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408715324.gif)
,垂足為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408731206.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408746256.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408778270.gif)
.
連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408793246.gif)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408809442.gif)
......................................................... (11分)
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408684442.gif" style="vertical-align:middle;" />,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408871265.gif)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408871281.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185408902405.gif)
為二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407717432.gif)
平面角的補(bǔ)角. ....................................... (12分)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409012433.gif)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409027725.gif)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409043408.gif)
.
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409074452.gif)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409090750.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185409105635.gif)
........................................................... (13分)
所以二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407717432.gif)
的余弦值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823185407732284.gif)
.......................................... (14分)