Express `2 hat i- hat j+3 hat k` as the sum of a vector parallel and a vector perpendicular to `2 hat i+4 hat j-2 hat kdot`

Explanation

Given that,
`vec a= 2 hat i- hat j+3 hat k`
`vec b=2 hat i+4 hat j-2 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 (2-1+3)-lambda (2+4-2)=0`
` implies lambda = (4)/4=1`.
`:. vec(a)=-vec(b)+vec(c) implies vec(c)= (vec(a)+vec(b))`.