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I'm a bit confused on where the logic/intuition for this approach comes from. What are x and c?


What is N here? I looks like a vector perpendicular to the line, but why the multiplication between it and x simulate a line? Is c a constant?


^ I think c is a constant and x is a point that satisfies the "nTx = c" But I am also a little confused about "nT", is nT just a vector parallel to the line? And how is t defined here?


Can this same formula be used to compute the closest on any hyper plane for R^n?


Could you also do this by writing p in the basis formed by the normal and the line as unit vectors?


Could you just write this in coordinated version so we can understand it more easily?


I understand that N is the normal to the line, but then N^T x is the dot product of N and the vector to some point x on the line, but then is this constant because we know that for some x and y on the line, N^T(y-x) = 0?


It seems like a linear equation to be solved, I guess the matrix could work here.


Could there possibly be an example of this? I do not quite understand what these abstract variables mean.