1. What is the Gram-Schmidt Process?

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space. It takes a finite, linearly independent set of vectors and generates an orthogonal set that spans the same subspace.

2. How Does the Calculator Work?

The calculator implements the Gram-Schmidt process:

Where:

• \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \) — Original input vectors
• \( \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 \) — Resulting orthogonal unit vectors
• \( \text{proj}_{\mathbf{u}}\mathbf{v} \) — Projection of vector v onto u
• \( \|\mathbf{v}\| \) — Norm (length) of vector v

3. Importance of Orthogonal Unit Vectors

Details: Orthogonal unit vectors are fundamental in linear algebra, physics, and engineering. They form bases for vector spaces, simplify calculations, and are essential in applications like computer graphics, signal processing, and quantum mechanics.

4. Using the Calculator

Tips: Enter three vectors as comma-separated values (e.g., "1,0,0"). All vectors must have the same dimension. The calculator will output three mutually orthogonal unit vectors.

5. Frequently Asked Questions (FAQ)

Q1: What if my vectors are linearly dependent?
A: The Gram-Schmidt process requires linearly independent vectors. If vectors are dependent, one of the resulting vectors will be zero.

Q2: Can I use this for more than 3 vectors?
A: This calculator handles 3 vectors, but the process extends to any number of vectors.

Q3: What's the difference between orthogonal and orthonormal?
A: Orthogonal vectors are perpendicular; orthonormal vectors are both perpendicular and unit length.

Q4: Why would I need orthogonal unit vectors?
A: They're useful for creating coordinate systems, solving systems of equations, and performing transformations.

Q5: Can I use this for 2D vectors?
A: Yes, just enter vectors with two components (e.g., "1,0").