Gram-Schmidt 3D

The Gram-Schmidt process turns three vectors $\vec v_1,\vec v_2,\vec v_3$ in $\mathbb{R}^3$, generally not orthogonal to each other, into an orthonormal basis $\vec u_1,\vec u_2,\vec u_3$. The tips of the vectors can be dragged in “Edit v₁ v₂ v₃” mode; the view rotates by dragging the background.

Compute & keepEdit v₁ v₂ v₃Resetslow motion1.00×

v₁, v₂, v₃ = —u₁, u₂, u₃ = —‖u₁‖, ‖u₂‖, ‖u₃‖ = —u₁·u₂, u₁·u₃, u₂·u₃ = —

The process, step by step

Step 1. The first unit vector is obtained by normalizing $\vec v_1$, i.e. dividing it by its own norm:

$\vec u_1=\dfrac{\vec v_1}{\lVert \vec v_1\rVert}$

Step 2. The component of $\vec v_2$ along $\vec u_1$ (its projection onto $\vec u_1$) is subtracted, leaving the orthogonal part $\vec w_2$; $\vec w_2$ is then normalized:

$\vec w_2=\vec v_2-(\vec v_2\cdot\vec u_1)\,\vec u_1,\qquad \vec u_2=\dfrac{\vec w_2}{\lVert \vec w_2\rVert}$

Step 3. Both projections are subtracted from $\vec v_3$, onto $\vec u_1$ and onto $\vec u_2$; the remaining orthogonal part $\vec w_3$ is then normalized:

$\vec w_3=\vec v_3-(\vec v_3\cdot\vec u_1)\,\vec u_1-(\vec v_3\cdot\vec u_2)\,\vec u_2,\qquad \vec u_3=\dfrac{\vec w_3}{\lVert \vec w_3\rVert}$

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 orthonormality

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 unit vectors $\vec u_1,\vec u_2,\vec u_3$, which appear one at a time during “Compute & keep”, in sync with the animation;
• the norms $\lVert\vec u_1\rVert,\lVert\vec u_2\rVert,\lVert\vec u_3\rVert$, which should stay at $1$ (the resulting vectors are unit vectors);
• the cross dot products $\vec u_1\cdot\vec u_2,\ \vec u_1\cdot\vec u_3,\ \vec u_2\cdot\vec u_3$, which should stay at $0$ (the resulting vectors are pairwise orthogonal).

Unit norms together with zero dot products are exactly the definition of an orthonormal basis: those live numbers offer a way to check, for any linearly independent starting triple, that the process worked correctly.

See also

For the variant that stops at orthogonality without normalizing the resulting vectors, see Gram-Schmidt 3D without normalization.

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 orthonormalization solver.