Gram-Schmidt Orthogonalization in 3D

The Gram-Schmidt process is one of the most-used algorithms in linear algebra: given any set of linearly independent vectors, it builds an orthonormal basis — vectors that are mutually perpendicular and exactly unit length — by projecting out the overlap between each new vector and the ones already processed. This simulator renders three real 3D vectors, lets you drag two of them freely, and shows the resulting orthonormal frame updating live alongside the projection arrows the algorithm actually subtracts. Live readouts track the orthogonality error (how close the output is to perfectly perpendicular), the determinant of the orthonormal matrix (always ±1), and the volume of the original parallelepiped — making concrete both what the algorithm does and what it deliberately throws away.