\mathbf a\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix}Variables
- a,b: 3D vectors
- result: the quantity produced after the stated inputs and conditions are applied
How to use this formula
Calculates the three-dimensional vector cross product.
Important notes
- The result is perpendicular to both vectors.
Quick example
i×j=k.
Applicability, worked calculation, and verification
Domain and applicability
Apply Cross Product Components only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- The result is perpendicular to both vectors.
- 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.
Do not use this formula when
- Do not use Cross Product Components when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Cross Product Components, check zero, negative, and extreme input values before relying on the result.
- When using Cross Product Components, 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\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For a=(1,0,0) and b=(0,1,0), a×b=(0,0,1).
- Compute the first component 0·0-0·1=0.
- Compute the second component 0·0-1·0=0.
- Compute the third component 1·1-0·0=1.
Independent verification
The result is perpendicular to both input vectors and has unit length.
Common mistakes
- Do not reorder factors or mix incompatible dimensions when using Cross Product Components; matrix operations are order-sensitive.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use Cross Product Components 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\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix}. - 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 Cross Product Components.
Verified against: NIST Digital Library of Mathematical Functions
Automated quality check: Passed the core formula indexing gate.
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 Cross Product Components used for?
Calculates the three-dimensional vector cross product.
Can I copy this formula as LaTeX?
Yes. Copy \mathbf a\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix} 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.