Question

已知|a|=2，|b|=1，a与b的夹角为丌/3,求向量2a+3b，与3a-b的夹角?

Answer 1

|a|=2，|b|=1.
a.b = |a||b|cos丌/3
a.b = 1
|2a+3b|^2
=(2a+3b).(2a+3b)
=4|a|^2+9|b|^2+ 12a.b
=16+9+12
=37
|3a-b|^2 = 9|a|^2+|b|^2 - 6a.b =36+1-6=31
let 2a+3b，与3a-b的夹角 = x
(2a+3b).(3a-b)= |2a+3b||3a-b|cosx
6|a|^2-3|b|^2+7a.b= √37√31cosx
24-3+7=√37√31cosx
cosx = 28/√1147
x = arccos(28/√1147)
2a+3b，与3a-b的夹角 = arccos(28/√1147)

Answer 2

可以用特殊值法呀！令a=(1,2),b=(1,0)

Answer 3

a与b的夹角为丌/3=60度
,|a|=2，得：a^2=4,|b|=1得：b^2=1
ab=|a||b|cos60=1
(2a+3b)^2=4a^2+12ab+9b^2=16+12+9=37
得：|2a+3b|=√37
(3a-b)^2=9a^2-6ab+b^2=36-6+1=31
得：|3a-b|=√31
(2a+3b)(3a-b)=6a^2+7ab-3b^2=24+7-3=27
(2a+3b)(3a-b)=|2a+3b||3a-b|cosx
cox=27/√1147