\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\thetaVariables
- a, b: vectors
- aᵢ, bᵢ: vector components
- θ: angle between vectors
How to use this formula
Combines two vectors into a scalar and relates their components to the angle between them.
Important notes
- Perpendicular nonzero vectors have dot product zero.
- The component and angle formulas describe the same operation.
Quick example
(1, 2) · (3, 4) = 11.
Applicability, worked calculation, and verification
Domain and applicability
Defined for equal-dimensional real vectors in a Euclidean inner-product space; the result is a scalar rather than a vector.
Units
- the result uses the product of the two vectors’ component units
Assumptions and domain checks
- Perpendicular nonzero vectors have dot product zero.
- For the Vector Dot Product, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.
- The angle unit and the domain or branch of any inverse trigonometric function must be explicit.
Do not use this formula when
- Do not use Vector Dot Product when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Vector Dot Product, check zero, negative, and extreme input values before relying on the result.
- When using Vector Dot Product, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For vectors (1, 2, 3) and (4, −1, 2), the dot product is 1·4 + 2·(−1) + 3·2 = 4 − 2 + 6 = 8.
- Confirm both vectors have the same number of components.
- Multiply corresponding components: 4, −2, and 6.
- Add the component products: 4 − 2 + 6.
- Report the scalar dot product as 8.
Independent verification
Using a·b = |a||b|cos θ gives the same scalar when the vector magnitudes and included angle are evaluated.
Common mistakes
- Do not omit the index or bounds in Vector Dot Product; changing either one changes which terms are included.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use Vector Dot Product in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Vector Dot Product.
Verified against: OpenStax Calculus Volume 1; NIST Digital Library of Mathematical Functions
Automated quality check: Passed the core formula indexing gate.
Formula references
- Calculus Volume 1OpenStax, Rice University — Reviewed limits, derivatives, integrals, and foundational calculus formulas.
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Vector Dot Product used for?
Combines two vectors into a scalar and relates their components to the angle between them.
Can I copy this formula as LaTeX?
Yes. Copy \mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.