Gram-Schmidt orthonormal basis for x1 + x2+ x3 =0

Question

I need to find an orthonormal basis for the plane x1 + x2 + x3 = 0 by using Gram-Schmidt on the vectors $$ \begin{pmatrix} 1 \\-1 \\0 \end{pmatrix}, \begin{pmatrix} 0 \\1 \\ -1 \end{pmatrix},\begin{pmatrix} 1 \\0 \\-1 \end{pmatrix}$$

I know how to use Gram-Schmidt on a basis that contains a few vectors, but I don't know what to do with that plane equation. Can anyone help me?

Answer 1

Any two of your vectors span the subspace but are not orthogonal. If you look at the Graham-Schmidt process you only subtract multiples of the earlier vectors at each stage. If all your vectors start out within a subspace, as they are here, the ones that come out of the process will also be within the subspace. So just do the process as normal on the any two of the given vectors and you will be fine.