Projections

Shine a light straight down on a vector and it casts a shadow onto another — the projection. That shadow is the closest point on the target line to the original vector. Projections are how a 3D scene flattens onto a screen, how least-squares fits the best line to noisy data, and how Gram-Schmidt peels off components. The geometry of 'drop a perpendicular' turns out to solve an extraordinary range of problems.