Projection of origin on a hyperplane defined by $a^Tx + b = 0$, as given here is $ab$. But when I workout, it is $a\lvert b \lvert$. (distance from the origin to $a^Tx + b = 0$, is $\lvert b \lvert$ (assuming $a$ to be a unit vector). Hence the vector is $a\lvert b \lvert$.
Also, the way they come up with solution is not straightforward. Anyone can explian to me ?. Also, is there a closed form for the projection of a vector $y$ on a hyperplane (other than $a\lvert a^Ty + b \lvert$ ?
It looks as if you have normalized so $a^Ta=1$. There are different ways to view orthogonal projections. One is through a parameter saying that $a$ is normal to your plane so the projection of the origin should be $ta$ for some real $t$. Inserting in the equation for the plane yields $t(a^T a)+b=t+b=0$ so $t=-b$ and the vector is $-b a$. Taking more generally a vector $y$ the projected vector should be $ta+y$ (again for some $t$) and then $a^T (ta + y)+b=t+a^T y+b=0$ from which you get the result $y-a(a^T y+b)$. Does this clarify?