|a|=√3,|b| =3,|c|=2√3
a+b+c = 0
(a+b+c).a = 0
|a|^2 + a.b + a.c = 0 (1)
similarly
(a+b+c).b =0
a.b + |b|^2+ b.c =0 (2)
(a+b+c).c =0
a.c + b.c + |c|^2 = 0 (3)
(1) - (3) + (2)
|a|^2 + a.b + a.c-(a.c + b.c + |c|^2) + a.b + |b|^2+ b.c =0
|a|^2 -|c|^2 + |b|^2 + 2a.b =0
3-12+9+ 2a.b =0
=> a.b =0