Gram-Schmidt orthogonalization in 2D

What you are seeing: two input vectors $v_1, v_2$ and the orthonormal output $u_1, u_2$ produced by one step of Gram-Schmidt: $u_1 = v_1/|v_1|$, then $u_2 = (v_2 - \langle v_2, u_1\rangle u_1) / |v_2 - \langle v_2, u_1\rangle u_1|$. The dashed cyan vector shows the projection $\langle v_2, u_1\rangle u_1$ being subtracted; the orange vector is the residual that becomes $u_2$ after normalization.

Drag the sliders for the angles and magnitudes of $v_1$ and $v_2$. The readout reports $|\langle u_1, u_2\rangle|$, which stays at machine zero ($10^{-16}$) regardless of input. Pushing $v_2$ collinear with $v_1$ collapses the residual to zero (linearly dependent inputs), which the readout flags.