estimate the position of a $ x in mathbb {R} ^ 2 $ knowing its distance to four beacons $ v_i $, $ i in {1, .., 4 } $, by the least squares, is finding the least squares solution to $ y = Ax $, or $ y in mathbb {R} ^ $ 4 is the vector of distances between $ x $ and $ k_i $ and $ A in mathbb {R} ^ {4×2} $ is the matrix with the line vectors $ a_i ^ T = frac {v_i ^ T} { | {v_i} |} $.

I have trouble understanding where the wording comes from $ y = Ax $ the problem comes from intuitively. What projections under the hood correct this formulation? What's going on geometrically?