试题分析:(Ⅰ)设向量
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
两两所成的角均为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=0或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337866491.png)
, 又|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|=1,|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
|=2,|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|=3.则当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=0时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=2,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=6,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=3,此时 |
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|
2=
2+
2+
2+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
=14+22=36,∴ |
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|=6;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337866491.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=-1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=-3,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
=|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338038430.png)
=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338943388.png)
,此时 |
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|
2=
2+
2+
2+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
=14-11=3,∴ |
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337710344.png)
.
(Ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=0,即|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|=6时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
的夹角显然为0; 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337804297.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337866491.png)
,即|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337710344.png)
时,∵ (
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
)·
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004338943388.png)
,且|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
|·|
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337710344.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004340113358.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
>=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004340206453.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337352311.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337383258.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337336296.png)
的夹角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824004337726482.png)
.
点评:熟练运用向量的运算及数量积的概念是解决此类求模和夹角的常用方法