Shows orthogonal projection geometrically (a vector v, a subspace line S, a sliding candidate u∈S, then the minimizing orthogonal projection with residual r ⟂ S) and connects it to least squares via the projection operator P = A(A^T A)^{-1}A^T, highlighting that for orthogonal projections P is symmetric and idempotent (P^2 = P).
Two-panel canvas: left panel renders R^2 geometry with a grid-snapped, blocky style; right panel renders P as a 2×2 matrix (using a 2×1 A so P = uu^T) and animates emphasis of P^2=P and P^T=P. Time is segmented into phases (0–1) using ease() to smoothly transition from sliding u to locking at proj_S(v) and then to operator/least-squares text.