Calculator Scope
Use this tool to evaluate vector pairs, compare norms, inspect angles, and verify classic inner product identities.
- Dot product and norms
- Orthogonality and angle
- Cauchy-Schwarz check
- Triangle inequality
- Polarization identity
- Parallelogram identity
- Optional linearity test
Enter Vector Data
Example Data Table
| Vector x | Vector y | Vector z | α | β | x · y | Angle | Projection factor | Linearity check |
|---|---|---|---|---|---|---|---|---|
| [1, 2, 3] | [4, -1, 2] | [0, 1, 1] | 2 | -1 | 8 | 62.1819° | 0.380952 | Pass, both sides equal 9 |
The example shows a non-orthogonal vector pair with a valid linearity verification and a positive Gram determinant.
Formula Used
Inner product
<x, y> = Σ(xᵢyᵢ)
Norm
||x|| = √<x, x>
Angle
cos(θ) = <x, y> / (||x|| ||y||)
Orthogonality
<x, y> = 0 implies the vectors are orthogonal.
Cauchy-Schwarz inequality
|<x, y>| ≤ ||x|| ||y||
Triangle inequality
||x + y|| ≤ ||x|| + ||y||
Polarization identity
<x, y> = (||x+y||² - ||x-y||²) / 4
Parallelogram identity
||x+y||² + ||x-y||² = 2||x||² + 2||y||²
Linearity test
<αx + βy, z> = α<x, z> + β<y, z>
How to Use This Calculator
- Enter vector x and vector y using commas, spaces, or separate lines.
- Keep both vectors in the same dimension.
- Optionally add vector z and scalars α and β for linearity testing.
- Press Submit and Verify to compute the proof checks.
- Read the result area above the form for summary values and proof status.
- Use the graph to compare vector components visually.
- Download the result table as CSV or PDF when needed.
Frequently Asked Questions
1. What does this calculator actually prove?
It numerically verifies core inner product identities for the vectors you enter. It supports learning and checking examples, but it is not a symbolic theorem prover.
2. Can I enter fractions or decimals?
Decimals work directly. Fractions should be converted first, such as entering 0.5 instead of 1/2, unless your server-side parser is expanded later.
3. What happens if vectors have different dimensions?
The page returns a validation error. Inner products, sums, and proof checks require matching dimensions for the participating vectors.
4. Why is the angle unavailable sometimes?
If one vector has zero norm, the angle formula divides by zero. In that case, the calculator safely marks the angle as unavailable.
5. What is the projection factor?
It is <x, y>/||y||². Multiplying y by that factor gives the vector projection of x onto y.
6. Why include the Gram determinant?
It gives another positivity-based check. For real inner product vectors, the two-vector Gram determinant should stay nonnegative.
7. Does the calculator support complex vectors?
This version is written for real vectors. Complex inner products need conjugation, which would require a more advanced parser and formula set.
8. What do the CSV and PDF files contain?
They export the computed summary values and proof verification results from the page, making it easier to review, share, or archive calculations.