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Gram-Schmidt builds an orthogonal basis by subtracting from each vector its projection onto the ones already processed, then normalising. Dependent vectors reduce to zero and are dropped.
Gram-Schmidt step
wₖ = vₖ − Σⱼ<ₖ (vₖ·uⱼ / uⱼ·uⱼ)·uⱼ, then normalise
The first normalises to (0.9486833, 0.3162278); the second loses its projection and becomes orthogonal to it.
It is a numerical check: a genuinely orthonormal set has all pairwise dot products zero to within rounding error.