Express the vector `vec(a)=5hat(i)-2hat(j)+5hat(k)` as sum of two vectors such that one is parallel to vector `vec(b)=3hat(i)+hat(k)` and other is perpendicular to `vec(b)`.

Answer

`vec(a)=vec(a_(1))+vec(a_(2))` where `vec(a_(1))=2(3hat(i)+hat(k))=6hat(i)+2hat(k)` and `vec(a_(2))=-hat(i)-2hat(j)+3hat(k)`