Orthogonal projection over a subspace

Question

I tried to solve the orthogonal projection problem below:

"Let's F = [(1,1,-1)], the orthogonal projection of (2,4,1) under the orthogonal subspace of F is:"

The formula I used was:

=> Proj V under U = (VU/UU) * U
=> V*U = (2,4,1) * (1,1,-1) = 2*1+4*1+(1*-1) = 2+4-1 = 5
=> U*U = (1,1,-1) * (1,1,-1) = 1*1+1*1+(-1*-1) = 1+1+1 = 3
=> (5/3)*(1,1-1)
=> (5/3, 5/3, -5/3)

The possible answers are:

A) (1,2,3)
B) (1/3, 7/3, 8/3)
C) (1/3, 2/3, 8/3)
D) (0, 0, 0)
E) (1, 1, 1)

Could someone spot the error and give a step by step solution?
Thanks!

Answer 1

It seems to me that you tried to compute the orthogonal projection of $(2,4,1)$ on $\bigl\langle(1,1,-1)\bigr\rangle$, whereas you should orthogonal projection on $\bigl\langle(1,1,-1)\bigr\rangle^\perp$.

By the way, the correct answer is B). You can do it as follows:

1. First, take a basis of $\bigl\langle(1,1,-1)\bigr\rangle$. I took $(1,0,1)$ and $(0,1,1)$.
2. Apply Gram-Schmidt to this basis in order to get an orthonormal basis. I got the basis $e_1=\frac1{\sqrt2}(1,0,1)$ and $e_2=\frac1{\sqrt6}(-1,2,1)$.
3. Finally, compute $\bigl((2,4,1).e_1\bigr)e_1+\bigl((2,4,1).e_2\bigr)e_2$.

Note that the final result will always be the same, no matter the choice of the basis.