Finding the Orthonormal Basis for a Given Subspace W

[goffinj]

Homework Statement

Find the orthogonal projection of the given vector on the given subspace W of the inner product space V?
V=R3, u = (2,1,3), and W = {(x,y,z): x + 3y - 2z = 0}
I don't understand how to find the orthonormal basis for W?

Homework Equations

I don't understand how to find the orthonormal basis for W?
I know once you have an orthonormal basis then you know that the projection is just
proj. = <u,v1>*v1 + <u,v2>*v2
where v1,v2 are the orthonormal vectors of the basis for W

The Attempt at a Solution

Since x + 3y -2z = 0 I took three vectors that are a solution to that system and then used the gram-schmidth to make them orthogonal, then normalized them, then used it to calculate the porjection but it came out wrong. It is supposed to be 1/17*(26 104) where 26 and 104 are in a column vector.
So basically, how to do you find the right basis for W?

 

[vela]

How do you end up with a two-dimensional vector when you're working in R³?

It would help if you showed us your actual work rather than just describing what you did.

 

[Outlined]

1. Find a basis for W (does it has to be orthonormal? do not think so), note dim(W) = 2
2. find a non-zero vector z orthogonal to the basis vectors of W
3. Solve cz + x = du, where x is a unknown vector in W and c and d are unknown scalars.
4. Now x is your projection.