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Find the line through the origin farthest to the points $z_j = (x_j,y_j )$, i.e. the line such that the sum of the square-distances between the points and their orthogonal projections onto the line is maximal.

$$z_1 = (1, 1)\quad z_2 = (1, −1)\quad z_3 = (−1, −1)$$

My understanding and approach is that we have to find a line farthest from the points with the constraint of going through the origin. I think this line will be perpendicular to the best fit line that passes through the origin. If that is correct, then I have to find that best fit line using projections and then find the perpendicular line. Am I on the right track? Would someone be able to provide me with a detailed solution of how to do this?

My professor mentioned the use of SVD in this problem, but I am not quite sure how this would work here.

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A line that goes through the origin can be defined by $\theta ,$ the angle it makes with the $x$ axis. I would just compute the distance from each point to the line as a function of $\theta ,$ square and sum them, take the derivative with respect to $\theta ,$ set to zero. I would guess from the symmetry that you want the line that goes through $(1,-1)$, or an angle of $-\frac \pi 4$, but haven't done it.

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  • $\begingroup$ Interesting approach. Do you know how SVD would apply here? $\endgroup$ – Madhav Nakar Dec 19 '17 at 15:39

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