Sorry if I wrong in the English terms..
I have V which is Internal product space. W = sp{(1,0,-1),(1,1,0)} is a base, subspaces of V. v = (-3,0,5) is in V.
I need to find the orthogonal projection of v on W. I don't understand what I do wrong.
my answer:
(1,0,-1) * (((-3,0,5)(1,0,-1))/((1,0,-1)(1,0,-1))) + (1,1,0) * (((-3,0,5)(1,1,0))/((1,1,0)(1,1,0))) =
(-11/2, -3/2, 8/2)
the right answer is (-11/3, 2/3, 13/3)
Yes, the answer is $\left(-\frac{11}{3},\frac{2}{3},\frac{13}{3}\right)$ indeed. The formula that you used works for orthgonal basis, but $\{(1,0,-1),(1,1,0)\}$ is not an orthogonal basis.
Apply Gram-Schmidt to $\{(1,0,-1),(1,1,0)\}$, and you will get $\left\{\left(\frac1{\sqrt2},0,-\frac1{\sqrt2}\right),\left(\frac1{\sqrt6},\frac2{\sqrt6},\frac1{\sqrt6}\right)\right\}$. Apply your formula to this basis, and you will get the right answer.