【答案】
分析:向量表示错误,请给修改,谢谢
将已知等式变形,两边平方;利用向量模的平方等于向量的平方及向量的数量积公式求出
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/0.png)
、
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/1.png)
两个向夹角
的余弦值,求出
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/2.png)
、
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/3.png)
的夹角,再由以
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/5.png)
为邻边的平行四边形为菱形,即可求得
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/6.png)
与
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/7.png)
的夹角.
解答:解:设
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/8.png)
、
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/9.png)
两个向量的夹角为θ,由
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/10.png)
,
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/11.png)
、
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/12.png)
、
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/13.png)
是单位向量,
两边平方可得 1+2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/14.png)
+1=1,即
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/15.png)
=-
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/16.png)
.
即 1×1×cosθ=-
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/17.png)
,∴θ=120°.
由题意可得,以
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/19.png)
为邻边的平行四边形为菱形,故
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/20.png)
与
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112249506702661/SYS201312021122495067026014_DA/21.png)
的夹角为60°.
故答案为 60°.
点评:本题考查要求两个向量的夹角关键要出现这两个向量的数量积,解决向量模的问题常采用将模平方转化为向量的平方,属于中档题.