Express the vector `vec(a)=(6hat(i)-3hat(j)-6hat(k))` as sum of two vectors such that one is parallel to the vector `vec(B)=(hat(i)+hat(j)+hat(k))` and the other is perpendicular to `vec(b)`.

Answer

`vec(a)=- (hat(i)+hat(j)+hat(k))+(7 hat(i)-2hat(j)-5hat(k))`

Explanation

Let `vec(a)=lambda vec(b)+vec(c)`, where `vec(c) bot vec(b)`. Then, `vec(c)=(vec(a)-lambda vec(b)) bot vec(b)`.
`:. (vec(a)-lambda vec(b)). vec(b)=0 implies (vec(a).vec(b))-lambda (vec(b).vec(b))=0`
`implies (6-3-6)-lambda (1+1+1)=0 implies lambda = (-3)/3=-1`.
`:. vec(a)=-vec(b)+vec(c) implies vec(c)= (vec(a)+vec(b))`.