1
分析:根据已知等式,平方得:
![](http://thumb.1010pic.com/pic5/latex/57929.png)
-4
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
+4
![](http://thumb.1010pic.com/pic5/latex/110314.png)
=4…(*),由向量
![](http://thumb.1010pic.com/pic5/latex/44.png)
,
![](http://thumb.1010pic.com/pic5/latex/45.png)
的夹角为60°,得
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
=
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
,代入(*)并化简整理,得4+2
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
=
![](http://thumb.1010pic.com/pic5/latex/57929.png)
+4
![](http://thumb.1010pic.com/pic5/latex/110314.png)
,再利用基本不等式得到
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
≤2,得到当且仅当
![](http://thumb.1010pic.com/pic5/latex/20368.png)
=2
![](http://thumb.1010pic.com/pic5/latex/20370.png)
时,
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
的最大值为1.
解答:∵
![](http://thumb.1010pic.com/pic5/latex/179323.png)
∴
![](http://thumb.1010pic.com/pic5/latex/590835.png)
=4,即
![](http://thumb.1010pic.com/pic5/latex/57929.png)
-4
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
+4
![](http://thumb.1010pic.com/pic5/latex/110314.png)
=4…(*)
∵向量
![](http://thumb.1010pic.com/pic5/latex/44.png)
,
![](http://thumb.1010pic.com/pic5/latex/45.png)
的夹角为60°,
∴
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
=
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
cos60°=
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
代入(*),得
![](http://thumb.1010pic.com/pic5/latex/57929.png)
-2
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
+4
![](http://thumb.1010pic.com/pic5/latex/110314.png)
=4,所以4+2
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
=
![](http://thumb.1010pic.com/pic5/latex/57929.png)
+4
![](http://thumb.1010pic.com/pic5/latex/110314.png)
≥4
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
解之得:
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
≤2,当且仅当
![](http://thumb.1010pic.com/pic5/latex/20368.png)
=2
![](http://thumb.1010pic.com/pic5/latex/20370.png)
时,等号成立
∴
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
=
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/20368.png)
![](http://thumb.1010pic.com/pic5/latex/20370.png)
,
![](http://thumb.1010pic.com/pic5/latex/44.png)
![](http://thumb.1010pic.com/pic5/latex/3695.png)
的最大值为1
故答案为:1
点评:本题给出向量等式,在已知两个向量夹角为60度的情况下,求它们数量积的最大值,着重考查了平面向量数量积的公式和基本不等式求最值等知识,属于中档题.