Date: Jan 22, 2013 1:05 AM
Author: William Elliot
Subject: Re: Generalizing Orthogonal Projection
On Mon, 21 Jan 2013, firstname.lastname@example.org wrote:
> Say we have a 1-D subspace L (i.e., a line thru the origin) in R^3 , and
> let q=(x,y,z) be a point in R^3 which is not on the line. Then I don't see
> how to project q orthogonally onto L; I can see how to project q into a
> 2-D subspace P : the projection of q into P is the ortho complement, and
> every vector in P is in the orthogonal complement of the ortho projected
> line. But, the same is not the case with q and L. Sorry for the rambling;
> my question is then actually:
> If L is a 1-D subspace of R^3, and q=(x,y,z) is a point not on L. Can we
> define the orthogonal projection of q into L, or do we need to have a plane
> P (as subspace) , to define an ortho projection of q?
Yes, the orthogonal projection of q onto L, is the orthogonal
projection of q onto L within the plain determined by q and L.