Question

已知｜向量a｜=根号3 ，｜向量b｜=3，｜向量c｜=2倍根号3 ，且 向量a+向量b+向量c=0向量，则 向量a·向量b

Answer 1

|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

Answer 2

|a|=√3，|b|=3，|c|=2√3
|a|^2=3，|b|^2=9，|c|^2=12
|a|^2+|b|^2=|c|^2
也就是说 向量a⊥向量b
那么向量a·向量b=0