Question

已知平面向量a,b,c满足|a|=1,|b|=2,|c|=3,且a,b,c两两所成的角相等,则|a+b+c|等于

Answer 1

a,b,c两两所成的角相等,
则b=a*(cos(2pi/3)+isin(2pi/3))*|b|/|a|
=2a*(cos(2pi/3)+isin(2pi/3)),
c=a*(cos(-2pi/3)+isin(-2pi/3))*|c|/|a|
=3a*(cos(-2pi/3)+isin(-2pi/3))
|a+b+c|
=|a+2a*(cos(2pi/3)+isin(2pi/3))+3a*(cos(-2pi/3)+isin(-2pi/3))|
=|a|*|1+2*(cos(2pi/3)+isin(2pi/3))+3*(cos(-2pi/3)+isin(-2pi/3))|
=|1-1+i3^(1/2)-3/2-i3*3^(1/2)/2|
=|-3/2+i(3^(1/2)-3/2*3^(1/2))|
=(9/4+3/4)^2
=3^(1/2)