LaTeX
\operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf bVariables
- a: vector being projected
- b: nonzero direction vector
How to use this formula
Projects one vector onto the direction of another.
Important notes
- b cannot be the zero vector.
Quick example
The result is parallel to b.
Applicability, worked calculation, and verification
Assumptions and domain checks
- b cannot be the zero vector.
- For the Vector Projection, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.
- Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold.
Worked example
The result is parallel to b.
Common mistakes
- When copying Vector Projection, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use Vector Projection 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
\operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- 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 Projection used for?
Projects one vector onto the direction of another.
Can I copy this formula as LaTeX?
Yes. Copy \operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b 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.