Date: Jan 22, 2013 3:21 AM
Author: Kaba
Subject: Re: Generalizing Orthogonal Projection

22.1.2013 7:47, hbertaz@gmail.com wrote:
> 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?


Let p, d, q in R^n, where p is a point on the line L, d is the direction
vector of L, and q is the point that is to be projected onto L.
Parametrize L by

f : R --> R^n: f(t) = p + td.

We want to find a t' such that

dot(f(t') - q, d) = 0,

where dot stands for an inner product. Using bilinearity,

dot(p + t'd - q, d) = 0
<=>
t' dot(d, d) + dot(p - q, d) = 0
<=>
t' = dot(q - p, d) / dot(d, d).

Therefore the orthogonal projection of q onto L is given by

f(t') = p + t'd.

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