orthonormal basis

An orthonormal basis is the gold-standard coordinate system: a set of basis vectors that are all mutually orthogonal (every pair meets at a right angle) and each exactly length 1. Ortho for the right angles, normal for the unit lengths.

When your basis is orthonormal, finding coordinates stops being work. To learn how much of a vector lies along one basis direction, you just take the dot product with that direction, no equation-solving required.

The everyday axes (1,0) and (0,1) are the simplest example. Stack an orthonormal basis as the columns of a matrix and you get an orthogonal matrix, a pure rotation or reflection that never stretches anything.