? +2????????? ，与向量2????????? +????????? 的夹角为?? 22.【解析】（1）设向量????????∴????????=
????? +2????????? )?(2????????? +????????? )(????
， ????? +2????????? |?|2????????? +????????? ||????
2
2
????? |=|????????? |=??，∴????????=2??+2??=4. 令|????
5???5??5
√√????? |=|????????? |=√2，∴|?????????? |=1， （2）∵|????
????? |=??，则|?????????? |=1???，而????? ????? =2?????????? ， 设|????????+????
????? ?????????? +????????? ?????????? =????????? ?(????????? +????????? )=2????????? ??????????? =2|????????? ||?????????? |???????? ∴????
1
=?2??(1???)=2???2??=2(???2)?2，
2
12
????? ?????????? +????????? ?????????? 的最小值是?. 当且仅当??=2时，????2（3）设∠??????=???∠??????=2???， ????? |=2， ????? ?????????? =2，????????? ?????????? =1，|????∵????
????? |????????=2?|????????? |=1， ∴2?|????????????
????? |??????(?????)=1?|????????? |=1， 同理：2?|????22????????
????? +????????? +????????? |=????????? 2+????????? 2+????????? 2+2????????? ?????????? +2????????? ?????????? +2????????? ?????????? ∴|????
2
??
11
?11??4?2?4 22cos?4sin?sin2??cos2?sin2??cos2????10 22cos?4sin?sin2?cos2?45sin2?cos2?454549 ????2??1??2222cos?4sin?4cos?4sin?444sin2?cos2?27
????? +????????? +????????? |=. 当且仅当时，所以|??????tan????????222cos?4sin?2
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