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Orthonormal Basis Generator

Generate orthonormal basis using Gram-Schmidt and normalization for 2D and 3D vectors

Enter 2D Vectors

Vector v1

Vector v2

Orthonormal Basis Formula

1. Gram-Schmidt: u1, u2, u3 (orthogonal)

2. Normalize: e1 = u1/||u1||, e2 = u2/||u2||, e3 = u3/||u3||

||u|| = sqrt(u·u) (Euclidean norm)

e·e = 1, e_i·e_j = 0 for i≠j

Orthonormal basis vectors are both orthogonal (dot product zero) and unit length (norm 1). First use Gram-Schmidt for orthogonal vectors, then normalize each by dividing by its norm. This is ideal for transformations, rotations, and simplifying matrix operations.

⚠ Input vectors must be linearly independent. If dependent, zero vectors will appear before normalization.

What is Orthonormal Basis?

Orthonormal basis is a special basis where vectors are both orthogonal and unit length. Orthogonal means dot product is zero (right angles), unit length means each vector has norm exactly 1. This combination has many nice properties that simplify calculations in linear algebra and geometry.

Unit Vector

||e|| = 1. Length exactly 1 unit. Divide u by ||u|| to normalize.

Orthogonal

e_i·e_j = 0 for i≠j. Vectors are perpendicular, no overlap.

Orthonormal

Both orthogonal + unit length. Best basis for computations.

Standard Basis

i=[1,0,0], j=[0,1,0], k=[0,0,1] are orthonormal by definition.

Teaching Example: v1=[1,0], v2=[1,1].
1. u1=[1,0], ||u1||=1 → e1=[1,0]
2. u2=[0,1], ||u2||=1 → e2=[0,1]
3. Check: e1·e2=0, ||e1||=||e2||=1 ✓

Applications

3D Rotations Computer Vision Signal Processing Quantum Mechanics Graphics

Frequently Asked Questions

What is orthonormal?▼

Orthogonal + unit length. u·v=0 and ||u||=||v||=1.

How to normalize?▼

Divide by norm: e = u / ||u||. ||e|| = 1.

Orthonormal vs orthogonal?▼

Orthonormal adds unit length constraint. Orthogonal just needs dot product zero.

Why orthonormal basis?▼

Simplifies matrix operations, rotations, and projections.

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