If with reference to the right handed system of mutually perpendicular unit vectors `hati,hatj` and `hatk,vec(alpha)=3hati-hatj,vec(beta)=2hati+hatj-3hatk`, then express `vec(beta)` in the form `vec(beta)=vec(beta)_(1)+vec(beta)_(2)`, where `vec(beta)_(1)` is parallel to `vec(alpha)` and `vec(beta)_(2)` is perpendicular to `vec(alpha)`.

Answer

`vec(beta)_(1)=(3)/(2)hati-(1)/(2)hatj` and `vec(beta)_(2)=(1)/(2)hati+(3)/(2)hatj-3hatk`