Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R^3 spanned by x and y?

Note: These vectors are in R^4.
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First, we use Gram-Schmidt to determine an orthogonal basis.

u = x = [-12, 9, -6, 0]
v = y - [(y · u)/(u · u)] u = [0, -3, 10, -2] - (-87/261) [-12, 9, -6, 0] = [-4, 0, 8, -2].

Check: [-12, 9, -6, 0] is orthogonal to [-4, 0, 8, -2], because their dot product is 0.

Finally, make u and v unit length.
==> [-12, 9, -6, 0]/√261 and [-4, 0, 8, -2]/√84 determine the orthonormal basis.

I hope this helps!