Given #hat(a_1),hat(a_2)and hat(a_3)# are unit vectors
So #abshat(a_1)=abshat(a_2)=abs hat(a_3)=1#
Let the angle between #hat(a_1) and hat(a_2)# is #theta#
Again given
#2hat(a_1)+2hat(a_2)+hat(a_3) =0.....[1]#
So #2abs(hat(a_1)+hat(a_2))=abshat(a_3) #
#=>2sqrt(abshat(a_1)^2+abshat(a_2)^2+2abshat(a_1)abshat(a_2)costheta)=abshat(a_3) #
#=>2sqrt(1^2+1^2+2*1*1*costheta)=1 #
#=>2sqrt(2+2costheta)=1 #
#=>8(1+costheta)=1 #
#=>costheta=1/8-1=-7/8 #
#=>cos^-1(-7/8)~~151^@#
Alternative way
We have
#2hat(a_1)+2hat(a_2)+hat(a_3) =0#
#=>2hat(a_1)*hat(a_1)+2hat(a_2)*hat(a_1)+hat(a_3)hat(a_1) =0#
#=>2+2abshat(a_2)abshat(a_1)costheta+hat(a_3)hat(a_1) =0#
#color(red)(=>2+2costheta+hat(a_3)hat(a_1) =0.......[2])#
similarly
#2hat(a_1)*hat(a_2)+2hat(a_2)*hat(a_2)+hat(a_3) *hat(a_2)=0#
#color(blue)(=>2costheta+2+hat(a_3) *hat(a_2)=0......[3])#
And also
#2hat(a_1)*hat(a_3)+2hat(a_2)*hat(a_3)+hat(a_3) *hat(a_3)=0#
#=>color(green)(2hat(a_1)*hat(a_3)+2hat(a_2)*hat(a_3)+1)=0.......[4])#
Combining [2],[3]and [4] we get
#8(1+costheta)=1 #
#=>costheta=1/8-1=-7/8 #
#=>cos^-1(-7/8)~~151^@#
Shortest method
#2hat(a_1)+2hat(a_2)=-hat(a_3) #
#=>(2hat(a_1)+2hat(a_2))*(2hat(a_1)+2hat(a_2))=hat(a_3) *hat(a_3) #
#=>4(1+2hat(a_2)*hat(a_1)+1)=1 #
#=>8(1+costheta)=1 #
#=>costheta=1/8-1=-7/8 #
#=>cos^-1(-7/8)~~151^@#