projection

The projection of a vector onto a line or a subspace is its shadow: imagine the sun straight overhead casting the vector down onto the line. The shadow is the point in that subspace that sits closest to the original vector.

Projecting splits a vector cleanly into two pieces: a part that lies inside the subspace (the shadow) and a leftover part that is perpendicular to it (the error, or residual). Those two pieces add back up to the original vector, and they are orthogonal to each other.

Onto a single direction u, the formula is proj = ((v . u) / (u . u)) * u. Example: projecting v=(3,4) onto the x-axis u=(1,0) gives (3,0) inside the axis, with leftover (0,4) standing straight up.