--- title: "Inner Products, Length and Orthogonality" description: "An inner product gives a real vector space length and angle. Orthogonal sets of nonzero vectors are independent. Inner products with an orthonormal basis give a vector's coordinates." canonical: https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W source: https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Inner Products, Length and Orthogonality An inner product gives a real vector space length and angle. Orthogonal sets of nonzero vectors are independent. Inner products with an orthonormal basis give a vector's coordinates. Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W ## Ready when they can - Verify the inner product axioms for a given rule - Show that an orthogonal set of nonzero vectors is independent - Compute coordinates in orthogonal bases using inner products, including the nonunit norm-squared denominator and the orthonormal special case - Compute dot products in R2 and R3, and compute lengths from a stated real inner product - Decide orthogonality of small vectors from a zero inner product ## Lesson: Length, angle, and right angles The dot product is the model inner product, written . In R squared, (a, b) dot (c, d) = ac + bd. In R to the n, multiply matching components and add all n products. For nonzero u and v, cos theta = / (length(u) length(v)). Vectors are orthogonal when their inner product is zero. The dot product of (3, -1) and (1, 3) is 3 - 3 = 0. A real inner product must be symmetric: = . It must be linear in the first input: = a + b. Symmetry gives linearity in the second input. Linearity in each input is called bilinearity. It must also be positive definite: >= 0, with equality only for v = 0. These rules must hold for all vectors and real scalars. One successful numerical check cannot prove them. An orthogonal set of nonzero vectors is independent. If c1 v1 + ... + ck vk = 0, take the inner product with vj. Orthogonality removes every term except cj. Positive definiteness gives > 0, so cj = 0. This works for every j. Thus a two-dimensional space can contain at most two mutually orthogonal nonzero vectors. *(drawing: The inner product gives squared length; take its square root to get length.)* An orthonormal basis consists of mutually orthogonal unit vectors. The coordinate along ej is . For e1 = (3/5, 4/5), e2 = (-4/5, 3/5), and v = (2, 1), the coordinates are 2 and -1. Both basis vectors have length one and their dot product is zero. A negative coordinate means the component points opposite to that basis vector. For an orthogonal basis that is not normalised, the coordinate along uj is / . This follows by taking the inner product of v = sum cj uj with uj. For an orthonormal basis the denominator is one. Coordinates depend on direction: if e2 = (0, -1), then <(a, b), e2> = -b, not b. Length in any real inner-product space is sqrt(). For the dot product, length(a, b) = sqrt(a squared + b squared). Other inner products can give other lengths. The rule = u1 v1 + 4 u2 v2 is symmetric and bilinear, and = v1 squared + 4 v2 squared is positive for every nonzero v. In contrast, u1 v1 alone gives zero squared length to (0, 1), so it is not positive definite. **Recap.** Test the axioms. Use dot products to find lengths and right angles, and read orthonormal coordinates directly from the inner product. *Lesson and assessment text authored for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX) ## Optional connections - [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw) ## Opens up - [Clustering: Structure Without Labels](https://lightmysky.com/learn/computing/clustering-structure-without-labels-mt_BAMGOlWgRY) - [Spacetime Intervals and Four-Vectors](https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf) - [Fourier Series](https://lightmysky.com/learn/mathematics/fourier-series-mt_DJ3iwfj7NK) - [Orthogonal Projection and Least Squares](https://lightmysky.com/learn/mathematics/orthogonal-projection-and-least-squares-mt_kGvRrYJrat) - [Lp Spaces and the Inequalities They Rest On](https://lightmysky.com/learn/mathematics/lp-spaces-and-the-inequalities-they-rest-on-mt_Qd8jU26qgE) - [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX)