If ` vec a , vec b ,a n d vec c` are unit vectors such that ` vec a+ vec b+ vec c=0,` then find the value of ` vec adot vec b+ vec bdot vec c+ vec cdot vec adot`

Options

A. 0
B. `-(1)/(2)`
C. `-(3)/(2)`
D. 2

Answer

C

Explanation

Given `|a|=|b|=|c|=1 and a+b+c=0`
We have ,`(a+b+c).(a+b+c)=0`
`implies a.(a+b+c)+b(a+b+c) +c(a+b+c)=0`
`implies a.a+a.b+a.c+b.a+b.b+b.c+c.a+c.b+c.c=0`
`implies |a|^(2)+|b|^(2)+|c|^(2)+2(a.b+b.c+c.a)=0`
`[ because a.a =|a|^(2)and a.b=b.a]`
`implies 1+1+1+2 (a.b+b.c+c.a)=0 [ because |a|=|b|=|c|=1]`
`implies 3+2(A.b+b.c+c.a)=0`
`implies a.b+b.c+c.a=-(3)/(2)`