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Gram-Schmidt Calculator

Vector Configuration

Set the number of vectors and their dimension. The calculator supports up to 10 vectors with dimension up to 10.

Enter Vectors

Enter the components of your 3 vectors, each with dimension 3.

Vector
x1
x2
x3
v1
v2
v3
v1 = [0.003.004.00]
v2 = [1.000.001.00]
v3 = [1.001.003.00]

Orthonormal Basis

Orthonormal Vectors (e₁, e₂, ...):

e1 = [0.0000, 0.6000, 0.8000]
e2 = [0.8575, -0.4116, 0.3087]
e3 = [-0.5145, -0.6860, 0.5145]

Orthogonal Vectors (u₁, u₂, ...):

u1 = [0.0000, 3.0000, 4.0000]
u2 = [1.0000, -0.4800, 0.3600]
u3 = [-0.1765, -0.2353, 0.1765]

Original Vectors:

v1 = [0.0000, 3.0000, 4.0000]
v2 = [1.0000, 0.0000, 1.0000]
v3 = [1.0000, 1.0000, 3.0000]

Step-by-Step Process

Starting with vectors: v1 = [0.0000, 3.0000, 4.0000] v2 = [1.0000, 0.0000, 1.0000] v3 = [1.0000, 1.0000, 3.0000]
--- Step 1: Processing v1 = [0.0000, 3.0000, 4.0000] ---
Orthogonal vector u1 = [0.0000, 3.0000, 4.0000]
Normalized vector e1 = [0.0000, 0.6000, 0.8000]
--- Step 2: Processing v2 = [1.0000, 0.0000, 1.0000] ---
Subtract projection of v2 onto u1:
Projection = [0.0000, 0.4800, 0.6400]
u2 = [1.0000, 0.0000, 1.0000] - [0.0000, 0.4800, 0.6400] = [1.0000, -0.4800, 0.3600]
Orthogonal vector u2 = [1.0000, -0.4800, 0.3600]
Normalized vector e2 = [0.8575, -0.4116, 0.3087]
--- Step 3: Processing v3 = [1.0000, 1.0000, 3.0000] ---
Subtract projection of v3 onto u1:
Projection = [0.0000, 1.8000, 2.4000]
u3 = [1.0000, 1.0000, 3.0000] - [0.0000, 1.8000, 2.4000] = [1.0000, -0.8000, 0.6000]
Subtract projection of v3 onto u2:
Projection = [1.1765, -0.5647, 0.4235]
u3 = [1.0000, -0.8000, 0.6000] - [1.1765, -0.5647, 0.4235] = [-0.1765, -0.2353, 0.1765]
Orthogonal vector u3 = [-0.1765, -0.2353, 0.1765]
Normalized vector e3 = [-0.5145, -0.6860, 0.5145]
--- Final Result ---
Orthogonal basis:
u1 = [0.0000, 3.0000, 4.0000]
u2 = [1.0000, -0.4800, 0.3600]
u3 = [-0.1765, -0.2353, 0.1765]
Orthonormal basis:
e1 = [0.0000, 0.6000, 0.8000]
e2 = [0.8575, -0.4116, 0.3087]
e3 = [-0.5145, -0.6860, 0.5145]

Verification

All orthonormal vectors have unit length (magnitude ≈ 1)
All pairs of orthonormal vectors are orthogonal (dot product ≈ 0)
The orthonormal set spans the same subspace as the original vectors

Gram-Schmidt Calculator FAQ

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