【答案】
分析:根据已知等式,平方得:
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/0.png)
-4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/1.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/2.png)
+4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/3.png)
=4…(*),由向量
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/4.png)
,
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/5.png)
的夹角为60°,得
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/6.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/7.png)
=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/8.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/9.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/10.png)
,代入(*)并化简整理,得4+2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/11.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/12.png)
=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/13.png)
+4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/14.png)
,再利用基本不等式得到
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/15.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/16.png)
≤2,得到当且仅当
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/17.png)
=2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/18.png)
时,
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/19.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/20.png)
的最大值为1.
解答:解:∵
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/21.png)
∴
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/22.png)
=4,即
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/23.png)
-4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/24.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/25.png)
+4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/26.png)
=4…(*)
∵向量
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/27.png)
,
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/28.png)
的夹角为60°,
∴
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/29.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/30.png)
=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/31.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/32.png)
cos60°=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/33.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/34.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/35.png)
代入(*),得
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/36.png)
-2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/37.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/38.png)
+4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/39.png)
=4,所以4+2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/40.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/41.png)
=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/42.png)
+4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/43.png)
≥4
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/44.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/45.png)
解之得:
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/46.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/47.png)
≤2,当且仅当
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/48.png)
=2
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/49.png)
时,等号成立
∴
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/50.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/51.png)
=
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/52.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/53.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/54.png)
,
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/55.png)
![](http://thumb2018.1010pic.com//pic6/res/gzsx/web/STSource/20131202112815816776400/SYS201312021128158167764016_DA/56.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.