22.1.2013 7:47, firstname.lastname@example.org 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,