Finding the Projection of a Vector onto a Subspace

[Dustinsfl]

Let S be a subspace of R³ spanned by u₂=\left[ \begin{array} {c}<br /> \frac{2}{3} \\<br /> \frac{2}{3} \\ <br /> \frac{1}{3} \end{array} \right] and u₃=\left[ \begin{array} {c}<br /> \frac{1}{\sqrt{2}} \\<br /> \frac{-1}{\sqrt{2}} \\ <br /> 0 \end{array} \right].
Let x=\left[ \begin{array} {c}<br /> 1 \\<br /> 2 \\ <br /> 2 \end{array} \right]. Find the projection of p of x onto S.

I know how to find projection but I am not sure about doing the projection on a subspace.

 

[lanedance]

how about finding an orthogonal basis of S (orthonormal is even better), then the projection of x onto each basis vector...?

 

[lanedance]

or equivalently, S represents a plane in \mathbb{R}^3, so find the projection of X on that plane...