Gram-Schmidt 3D without normalization

This is the variant of the Gram-Schmidt process that stops at orthogonalization: starting from three vectors $\vec v_1,\vec v_2,\vec v_3$ in $\mathbb{R}^3$, it produces three mutually orthogonal vectors $\vec w_1,\vec w_2,\vec w_3$ that are not of length $1$ (the final normalization step is missing). The tips of the vectors can be dragged in “Edit v₁ v₂ v₃” mode.

Compute & keepEdit v₁ v₂ v₃Resetslow motion1.00×

v₁, v₂, v₃ = —w₁, w₂, w₃ = —‖w₁‖, ‖w₂‖, ‖w₃‖ = — (not necessarily 1)w₁·w₂, w₁·w₃, w₂·w₃ = —

The process, step by step

Step 1. The first resulting vector coincides with $\vec v_1$: nothing needs to be subtracted or normalized.

$\vec w_1=\vec v_1$

Step 2. The projection of $\vec v_2$ onto $\vec w_1$ is subtracted. Since $\vec w_1$ is not a unit vector, the projection coefficient is $\dfrac{\vec v_2\cdot\vec w_1}{\vec w_1\cdot\vec w_1}$ (the dot product alone is not enough, as it would be with a unit vector):

$\vec w_2=\vec v_2-\dfrac{\vec v_2\cdot\vec w_1}{\vec w_1\cdot\vec w_1}\,\vec w_1$

Step 3. Both projections are subtracted from $\vec v_3$, onto $\vec w_1$ and onto $\vec w_2$, using the same kind of coefficient normalized by the squared norm of the vector being projected onto:

$\vec w_3=\vec v_3-\dfrac{\vec v_3\cdot\vec w_1}{\vec w_1\cdot\vec w_1}\,\vec w_1-\dfrac{\vec v_3\cdot\vec w_2}{\vec w_2\cdot\vec w_2}\,\vec w_2$

Pressing “Compute & keep” animates this sequence on the simulator, color by color, in the same order described above. The “slow motion” slider allows following it at a gentler pace, without changing the final result.

Why the projections flip

In the animation, projections don’t just fade away as the orthogonal vector grows: they are first flipped by $180°$ around the origin (giving their opposite, since subtracting a vector is the same as adding its opposite), then translated tip-to-tail onto the tip of the starting vector (in step 3, the second flipped projection is chained onto the tip of the first). The orthogonal vector then grows from the origin to the tip of this chain. This is a deliberate teaching choice: vectors are handled only in additive, tip-to-tail logic. Vector subtraction is carried out this way: by adding the opposite of the vector being removed.

Checking orthogonality

Below the simulator’s buttons, four numeric rows are always kept up to date:

• the coordinates of the starting vectors $\vec v_1,\vec v_2,\vec v_3$ (also editable by typing them directly, in “Edit v₁ v₂ v₃” mode);
• the coordinates of the resulting vectors $\vec w_1,\vec w_2,\vec w_3$, which appear one at a time during “Compute & keep”, in sync with the animation;
• the norms $\lVert\vec w_1\rVert,\lVert\vec w_2\rVert,\lVert\vec w_3\rVert$, marked as “not necessarily 1” because, unlike the orthonormalized version, they are not generally expected to equal $1$: the length of the resulting vectors depends on the starting vectors (for some particular starting triples they may still turn out to be $1$: that is a special case, not the rule);
• the cross dot products $\vec w_1\cdot\vec w_2,\ \vec w_1\cdot\vec w_3,\ \vec w_2\cdot\vec w_3$, which should instead stay at $0$ (the resulting vectors are pairwise orthogonal, even though not unit length).

A zero dot product is, on its own, the defining property of orthogonality: that live number offers a way to check, for any linearly independent starting triple, that the process worked correctly, regardless of the length of the resulting vectors.

See also

For the full variant, which also normalizes the resulting vectors into an orthonormal basis, see Gram-Schmidt 3D.

See also

This simulator only works in $\mathbb{R}^3$ and shows results as approximate decimal numbers. For the exact solution (in rational arithmetic, with fractions instead of decimals) and to work with more than three vectors or dimensions, use the Gram-Schmidt orthogonalization solver.