Calculator Inputs
Example Data Table
| Vector A | Vector B | |a·b| | ||a||·||b|| | Holds? |
|---|---|---|---|---|
| 1, 2, 3 | 4, 5, 6 | 32 | 32.83291 | Yes |
| 2, -1, 0 | -4, 2, 0 | 10 | 10 | Yes |
| 3, 0, 4 | 0, 5, 0 | 0 | 25 | Yes |
Formula Used
The Cauchy–Schwarz inequality states that for vectors a and b in a real inner-product space:
|⟨a,b⟩| ≤ ||a|| · ||b||
With optional positive weights, the inner product becomes ⟨a,b⟩ = Σ wᵢ aᵢ bᵢ and the norm becomes ||a|| = √(Σ wᵢ aᵢ²).
How to Use This Calculator
- Set the dimension n of your vectors.
- Enter exactly n numbers for Vector A and Vector B.
- Optionally enable weights and enter n positive weights.
- Select a proof method and your display precision.
- Press Compute & Show Proof to see results above the form.
- Use Download CSV or Download PDF to export.
Why this inequality supports reliable calculations
Cauchy–Schwarz bounds correlation between two numeric vectors and prevents impossible combinations of dot products and norms. In analytics, it explains why a normalized similarity score must stay within −1 and 1. For example, if ||a||=10 and ||b||=3, then |⟨a,b⟩| can never exceed 30, regardless of dimension.
Inner products and configurable weights
This calculator evaluates ⟨a,b⟩ and ||a||, ||b|| using either the standard dot product or a positive weight vector w. With weights enabled, each component is effectively scaled by √wᵢ, so the same inequality holds in a reweighted geometry. This is common in weighted least squares, portfolio variance, and sensor fusion where measurements have different reliability.
Discriminant proof and numeric sanity checks
The quadratic method studies f(t)=||a−tb||². Because f(t) is never negative, its discriminant must be non‑positive, yielding (⟨a,b⟩)² ≤ ||a||²||b||². The tool reports RHS−LHS and the ratio LHS/RHS to show tightness. A small tolerance (about 1e−12 scaled) is used to avoid false “violations” from rounding.
Angle interpretation and projection diagnostics
For nonzero vectors, cos(θ)=⟨a,b⟩/(||a||||b||). The calculator clamps cos(θ) into [−1,1] before computing θ in degrees, improving stability near the endpoints. The accompanying Plotly chart draws a, b, and the projection of a onto b (first two components), helping you see how alignment increases the dot product.
Equality cases and edge conditions
Equality occurs when a is a scalar multiple of b, including the zero‑vector case. The calculator tests proportionality with a small relative tolerance and shows an estimated scalar k when available. If ||b||=0, the RHS becomes 0 and the inner product must be 0; if both are zero, every proof method collapses cleanly.
Reporting, reproducibility, and practical limits
Inputs support up to n=20 components, with precision choices from 0 to 12 decimals for display. The example table illustrates typical outcomes and confirms that LHS never exceeds RHS. CSV export stores inputs, norms, inner product, and angle. PDF export adds the selected proof steps for audit, homework checking, and model documentation. Because exports are deterministic, teams can compare runs across revisions. Store CSV in QA folders, and attach the PDF to tickets. When investigating anomalies, check the ratio close to 1 and the reported angle near 0° or 180° for fast root-cause triage.
FAQs
Q1. What does the calculator actually verify?
A. It computes |⟨a,b⟩| and ||a||·||b||, then confirms the inequality numerically. It also generates proof steps consistent with the selected method and your entered vectors.
Q2. Why can the angle show “clamped” behavior?
A. Floating-point rounding can push cos(θ) slightly above 1 or below −1. Clamping keeps cos(θ) inside [−1,1] so the angle remains real and stable.
Q3. How do weights change the meaning of the result?
A. Weights define a new inner product Σ wᵢ aᵢ bᵢ and norm √(Σ wᵢ aᵢ²). The inequality still holds, but geometry is measured in the weighted metric.
Q4. When does equality occur in practice?
A. Equality holds when one vector is a scalar multiple of the other, or when at least one vector is the zero vector. The tool estimates k when it detects proportional components.
Q5. Why might the ratio LHS/RHS be close to 1?
A. A ratio near 1 indicates vectors are nearly collinear, making the bound tight. A ratio near 0 indicates near-orthogonality or very small dot product relative to norms.
Q6. What does the Plotly graph show for higher dimensions?
A. The chart visualizes the first two components (or a 1D fallback) and the projection of a onto b in that view. Higher-dimensional behavior is still reflected in the numeric checks.