Let `vec(a), vec(b) and vec(c) ` be three vectors such that `vec(a) + vec(b) + vec(c) = 0 and |vec(a)|=10, |vec(b)|=6 and |vec(c) |=14`.
What is `vec(a). vec(b) + vec(b).vec(c)+ vec(c). vec(a)`. equal to ?

Options

A. -332
B. -166
C. 0
D. 166

Answer

B

Explanation

We have `vec(a)+vec(b)+vec(c)=vec(0)`
So `|vec(a)+vec(b)+vec(c)|=0`
`rArr |vec(a)+vec(b)+vec(c)|^(2)=|vec(a)|^(2)+|vec(c)|^(2)+2(vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a))`
`rArr 0=(10)^(2)+(6)^(2)+(14)^(2)+2(vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a))`
`rArr 0=100+36+196+2(vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a))`
`rArr -(332)/(2)=vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a)`
`rArr -166=vec(a).vec(b)+vec(b).vec(c)+vec(c).vec(a)`