Decompose the vector `6 hat i-3 hat j-6 hat k` into vectors which are parallel and erpendicular to the vecrtor ` hat i+ hat j+ hat kdot`

Explanation

Given that,
`vec a=6 hat i-3 hat j-6 hat k`
`vec b= hat i+ hat j+ hat kdot`
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))`.