English: The animation illustrates the Gram-Schmidt process for obtaining an orthonormal basis of vectors, in this particular case, in 3-dimensional euclidean space.

1. The first of the original vectors, v₁, is taken as the first vector of the orthogonal basis, u₁.
2. The second of the original vectors, v₂, is then projected onto u₁. The projection is subtracted from v₂, obtaining a new vector u₂, which is orthogonal to u₁
3. The third of the original vectors, v₃, is projected onto u₁. The result is subtracted from v₃, obtained the intermediate vector v'₃. This intermediate vector was used for illustration purposes, in order to show the orthogonalization against each individual vector one at a time. Next, this intermediate vector is projected onto u₂, and the resulting vector is subtracted from v'₃, resulting in the vector u₃, which is orthogonal to both u₁ and u₂. At this point, we have an orthogonal basis, but not an orthonormal basis, as the vectors are not unit vectors.
4. In the last step, the vectors are scaled by the inverse of their norms, a process called normalization. The resulting vectors, e₁, e₂ and e₃ are mutually orthogonal, and are all unit vectors. Therefore, they form an orthonormal basis for the 3-dimensional euclidean space R³. This completes the process.