Question

设向量abc都是单位向量且a+b+c=0求a*b

Answer 1

∵向量abc都是单位向量
∴|a|=|b|=|c|=1
a+b+c=0 平方一下
a^2+b^2+c^2+2ab+2bc+2ac=0
|a|^2+|b|^2+|c|^2+2ab+2bc+2ac=0
3+2ab+2bc+2ac=0
3+2ab+2c(a+b)=0
∵a+b=-c
∴3+2ab+2c(-c)=0
3+2ab-2c^2=0
3+2ab-2=0
ab=-1/2

Answer 2

a+b+c=0
(a+b+c).a=0
=> a.b + a.c+1= 0 (1)

and
(a+b+c).b=0

=>a.b+b.c +1 =0 (2)

and
(a+b+c).c=0
=>a.c+b.c +1 =0 (3)

(2)-(1)
a.b -a.c =0 (4)

(1) +(4)
2a.b =-1
a.b =-1/2