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Two vectors are nonorthogonal when their inner product is nonzero. In the power method, the starting vector must be nonorthogonal to a dominant eigenvector.

Equip the function spaces of the two nonempty parts of a bipartite graph with uniform inner products. Its normalized adjacency operator is $({\theta }_{G}v)(x)={\mathbb{E}}_{y}G(x,y)v(y)$. In orthonormal coordinate bases, its matrix is $G/\sqrt{|X||Y|}$, not the unscaled adjacency matrix. For a biregular graph of edge density of a bipartite graph $\gamma$, constant functions give a largest singular value $\gamma$, and the remaining singular values belong to the centered kernel $G-\gamma$.

An N-term approximation in an orthonormal basis uses at most $N$ basis functions. For a chosen index set $\Lambda$, its optimal coefficients are the inner products with those functions, and its squared error is the sum of the omitted squared coefficients by the Parseval identity.

For a regular self-adjoint Sturm-Liouville problem with positive weight $w$ and self-adjoint boundary conditions, eigenfunctions with distinct eigenvalues are orthogonal in the weighted inner product $\int w\overline{u}v$. Integration by parts proves this by equating $\int \overline{u}Lv$ and $\int \overline{Lu}v$.

At equally replicated quantitative treatment levels, evaluations of orthogonal polynomials generate pairwise orthogonal treatment contrasts, using the replication-weighted inner product. Five equally spaced levels give four contrast directions representing degrees one through four, besides the constant direction. This decomposes treatment variation without assuming the response has low polynomial degree.

Two vectors are orthogonal when their inner product is zero.

For a metric connection on a vector bundle, the covariant derivative along a curve satisfies $(d/dt)⟨u,v⟩=⟨D_{t}u,v⟩+⟨u,D_{t}v⟩$. If both sections are parallel, the right side vanishes. Thus parallel transport preserves the fibre inner product, not merely vector lengths, and is a linear isometry between the endpoint fibres.