Watch the Gram-Schmidt orthogonalization process step by step in 2D and 3D with animated projections.
Feed in a set of vectors in 2D or 3D and the tool replays the Gram-Schmidt orthogonalization process as an animation: each vector is projected onto the span of the vectors processed before it, the projection is subtracted, and the perpendicular leftover is normalized to unit length. Every step shows the original vectors, the projection arrow, and the growing orthonormal basis. The most common surprise is that the output depends on the order you enter the vectors: the same set in a different order yields a different orthonormal basis, and a nearly dependent vector produces a tiny leftover that gets stretched into a full-size unit vector.
A concrete input and expected output from the current implementation.
Input
v1 = (3, 4), v2 = (1, 2) [2D]
Expected output
u1 = (0.6, 0.8), u2 = (-0.8, 0.6); the step-2 projection of v2 onto u1 is shown as the vector (1.32, 1.76), which has length 2.2, and the QR factorization is R = [[5, 2.2], [0, 0.4]]
The first vector is normalized to unit length: (3, 4) / 5 = (0.6, 0.8). The projection of (1, 2) onto u1 has length 2.2; subtracting it leaves (-0.32, 0.24), whose norm is 0.4, and normalizing gives (-0.8, 0.6). The two results are unit vectors and perpendicular, since 0.6 * -0.8 + 0.8 * 0.6 = 0.
Vectors are handled left to right, in the order entered. The first is scaled to unit length and becomes the first basis vector. Each later vector is projected onto the span of the basis vectors built so far, the projection is subtracted off, and the perpendicular remainder is normalized. If that remainder is zero, the vector is a linear combination of earlier ones and drops out of the basis.
Each step is an inner-product pass: the current vector is dotted with every finished basis vector, each basis vector is scaled by its dot product, and the scaled vectors are subtracted from the current one. Because the finished basis is orthonormal, the subtractions are independent and each one removes exactly one direction. The same numbers form a QR factorization: the orthonormal basis vectors are the columns of Q, and the lengths and inner products computed at each step are the entries of R.
No. Each vector is replaced by its part perpendicular to everything processed before it, then stretched to unit length. The projection being subtracted is often much longer than the leftover, so the direction changes dramatically; only the first vector keeps its original direction. What stays fixed is the span of the whole set, not the shape of individual vectors.
Indirectly, by watching what happens at each step. If a vector's leftover is zero, it was a combination of the earlier vectors and is dropped, so the output has fewer vectors than the input. In 2D that means all entered vectors lie on a single line; in 3D it means they lie in a line or a plane rather than filling the space.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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