I would like to know if my problem is correct. It is a homework from my university course.
The questions are:
a) Find the projection of $y$ on $x$
b) Find the vector projection of $y$ on $x$
c) Find the vector projection of $y$ on the normal of $x$
where $x = [5,2,-1]$, $y = [0,1,1]$
a)
$P = \frac{[0,1,1] \cdot [5,2,-1]}{\sqrt{30}} = √30 / 30 *(-5) = -3√30 / 18$
b)
$yx = -3√30 / 18 * √30 / 30 [5,2,-1] = -3 / 18 [5,2,-1] = [-15/18,-6/18,3/18]$
c)
$y \perp x = [0,1,1] - [-15/18,-6/18,3/18] = [15/18,4/3,15/18]$