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.