Verify dot products, angles, and norms in seconds. Supports custom metric matrices and weighted integrals. Get pairwise checks, clean output, and downloadable reports instantly.
| Case | Input | Inner product | Orthogonal? |
|---|---|---|---|
| Vector pair | v=(1,2,-1), w=(2,-1,0) | ⟨v,w⟩ = 0 | Yes |
| Vector set | [1,0,0], [0,1,0], [0,0,1] | Off-diagonals = 0 | Yes |
| Function pair | f=sin(x), g=cos(x), [0,π], w=1 | ∫ sin(x)cos(x) dx = 0 | Yes |
These are common reference examples; your numeric result may be near zero depending on tolerance and settings.
Two vectors are orthogonal when their inner product is zero:
Standard: ⟨v,w⟩ = v · w = Σ vᵢwᵢ
Metric: ⟨v,w⟩ = vᵀGw
On an interval [a,b], functions are orthogonal under a weight w(x) if:
⟨f,g⟩₍w₎ = ∫ₐᵇ f(x)g(x)w(x) dx = 0
This page approximates the integral using Simpson’s rule with an even number of subintervals.
Numerical tolerance ε is used because floating-point arithmetic rarely returns exact zeros for non-trivial inputs.
Orthogonality is tested by checking whether the inner product is close to zero. This calculator uses a user-set tolerance ε, so the rule is |⟨a,b⟩| ≤ ε. For vectors, ⟨v,w⟩ is either the dot product Σvᵢwᵢ or a metric form vᵀGw. For functions, ⟨f,g⟩₍w₎ is a weighted integral over [a,b].
Along with ⟨v,w⟩, the calculator reports ||v|| and ||w||, plus the angle θ computed from cos(θ)=⟨v,w⟩/(||v||·||w||). When ε is small (for example 1e−9), a near-zero inner product strongly indicates perpendicular directions in Euclidean space. For practical datasets, increasing ε can reduce false “not orthogonal” flags caused by rounding.
Selecting a metric matrix G changes the geometry: norms become ||v||₍G₎=√(vᵀGv) and orthogonality becomes vᵀGw≈0. This is useful in quadratic forms, optimization, and coordinate transforms. A symmetric G is recommended because it preserves a consistent inner-product structure. If G is diagonal, each component is scaled, letting you model anisotropic weighting of coordinates.
For multiple vectors, pairwise orthogonality is summarized by the Gram matrix VᵀV. Off-diagonal entries measure cross-correlation between distinct vectors, while diagonal entries are squared norms. The heatmap highlights near-zero off-diagonals quickly, which is helpful when m is large and manual checking is slow. Computational cost grows roughly with m²·n for m vectors of dimension n.
Function orthogonality is evaluated numerically using Simpson’s rule with an even number of subintervals. Increasing the subinterval count improves accuracy for oscillatory functions such as sin(kx) and cos(kx), especially on longer intervals. The graph also plots the weighted product f(x)g(x)w(x), making it easier to see positive and negative regions that cancel in the integral.
CSV export stores the key values used in the decision: tolerance, inner product, norms, angle, and pass/fail flags. PDF export provides a compact snapshot suitable for assignments, reviews, or audits. For consistent comparisons across runs, keep ε and numeric settings fixed and only vary inputs.
Floating-point calculations rarely return exact zeros. ε defines how close to zero an inner product must be to count as orthogonal, avoiding misleading results from tiny rounding errors.
Use a symmetric, dimension-matched matrix that represents your weighting or geometry. Diagonal matrices apply per-coordinate weights, while full matrices can encode correlated coordinate systems.
Yes. Orthogonal means the inner product is near zero. Orthonormal additionally requires each norm to be near one, so scaling a vector keeps orthogonality but breaks orthonormality.
The Gram matrix entries are dot products between every pair of vectors. Off-diagonal values close to zero indicate pairwise orthogonality, while diagonal values show each vector’s squared length.
Start with 200 for smooth functions. Increase for sharp changes or high-frequency oscillations. The value must be even; larger values improve accuracy but increase computation time.
Numerical integration error, an unsuitable interval, or an unexpected weight function can shift the integral away from zero. Increase subintervals, verify [a,b], and check w(x) for sign changes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.