Thread: Find the orthogonal projection

Find the orthogonal projection of (1 1 1 1) onto <(1 3 1 1), (2 -1 1 0)>.

I have no problem finding the orthogonal projection of a vector onto another vector, which is (<y, x> x)/<x, x>. However, I don't know how to find the orthogonal projection of a vector onto a scalar. Is there a way to do this?

Originally Posted by letitbemww

Find the orthogonal projection of (1 1 1 1) onto <(1 3 1 1), (2 -1 1 0)>.

I have no problem finding the orthogonal projection of a vector onto another vector, which is (<y, x> x)/<x, x>. However, I don't know how to find the orthogonal projection of a vector onto a scalar. Is there a way to do this?

There is no way to project a vector orthogonally onto a scalar. I think that you are misinterpreting the notation here. The angled brackets in <(1 3 1 1), (2 -1 1 0)> cannot mean an inner product. They must be intended to indicate the subspace spanned by the vectors (1 3 1 1) and (2 -1 1 0).

Oh that makes more sense. But wouldn't the subspace spanned by these vectors be the same vectors since they are linearly independent? So then would I just have to find the orthogonal projection onto each one?

Originally Posted by letitbemww

Oh that makes more sense. But wouldn't the subspace spanned by these vectors be the same vectors since they are linearly independent? So then would I just have to find the orthogonal projection onto each one?

To find the projection onto a two-dimensional subspace, you need to add the projections onto two orthogonal vectors in that subspace. The vectors (1 3 1 1) and (2 -1 1 0) are linearly independent but they are not orthogonal.