Orthogonality

Two vectors at right angles don't interfere — push north and you don't move east. That independence is the dot product's zero: a·b = 0 means perpendicular, no shared direction. An orthogonal basis, where every pair is at right angles, makes coordinates trivial (no cross-contamination) and computations stable. When life hands you a slanted basis, Gram-Schmidt is the recipe that straightens it into an orthogonal one.