Express as the sum of a vector parallel and a vector perpendicular to

Let and

let where u is vector parallel to b and v is vector perpendicular to b.

So, ; where p is some constant

Substituting this value in above equation

Now according to conditions since vector v and b are perpendicular to each other

2(2 – 2p) – 4(1 + 4p) – 2(3 + 2p) = 0

4 – 4p – 4 – 16 p – 6 – 4p = 0

– 24 p = 6

Substituting this value of u vector in above equation