topic Re: Interpreting Vector of Residuals in Statistical Procedures

Question

I've been reading the Projection section of this wiki article on ordinary least squares: https://en.wikipedia.org/wiki/Ordinary_least_squares#Normal_matrix

But I'm not sure how to make sense of the vector of residuals. I'm using a very simple example on purpose just so it's easier to understand.

In any case...

proc iml;
A = {4 8,
     6 12};
b = {5, 1};

I had x of 0.1 and 0.2, p (projection) column vector [2, 3] and resulting vector residual of column vector [3, -2] ...but I have no idea how to interpret it beyond understanding that the residual is apparently the shortest of all possible vectors.

So, first, is the residual also referred to as..

a. the orthogonal complement of b on to C(A)?

b. How can I visualize the residual.

How do I make sense of what a coordinate [3, -2] means for a vector residual?