Quick answer
The cross product of 3D vectors is a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) the result is (−3, 6, −3), with magnitude 7.3485.
Enter two 3-component vectors. The calculator returns the cross product and its magnitude.
Related tools: Dot Product Calculator: Dot product of two vectors. Solve for Unknown Fraction (Cross Multiplication): Solve a/b = c/d when one value is unknown. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Cross Product Calculator
- Type vector A, for example 1, 2, 3.
- Type vector B, for example 1, 2, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Vector A | Text, such as 1, 2, 3 | — |
| Vector B | Text, such as 1, 2, 3 | — |
Formula and method
Formula
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
IEEE-754 double precision; shown to at most 6 decimal places.
Defined for three components.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Vector A = 1, 2, 3; Vector B = 4, 5, 6.
| Result | Value |
|---|---|
| Cross product | (-3, 6, -3) |
| Magnitude | 7.348469 |
Example 2
Inputs: Vector A = 1, 0, 0; Vector B = 0, 1, 0.
| Result | Value |
|---|---|
| Cross product | (0, 0, 1) |
| Magnitude | 1 |
The first line is the headline result (cross product). Below it the cross Product Calculator lists magnitude, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Meaning
The cross product is perpendicular to both vectors, and its magnitude equals the area of the parallelogram they span: |a||b| sin θ. Here √(9 + 36 + 9) = √54 = 7.3485.
Order matters
a × b = −(b × a), so swapping the vectors flips the sign. Parallel vectors give the zero vector. Exactly 3 components are needed; the 2D equivalent is a single number.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Vector A: entering "1, 2" shows "The cross product needs exactly 3 components." beside that field and no result.
Choosing between the Cross Product Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Dot Product Calculator](/math/dot-product-calculator/) | vector A and vector B | Dot product |
| [Solve for Unknown Fraction (Cross Multiplication)](/math/solve-for-unknown-fraction/) | Unknown (d) | |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Exactly 3 components
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for vector A to confirm this.
Frequently asked questions
What is the cross product of parallel vectors?
The cross product of parallel vectors is the zero vector, because sin 0° = 0.
What is the right-hand rule?
Curl fingers from a to b; the thumb gives the direction.
Where is it used?
Torque, angular momentum and normals to surfaces.
What does the Cross Product Calculator calculate?
Cross product of two 3-D vectors. It returns Cross product and Magnitude.
What formula does the Cross Product Calculator use?
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Defined for three components.
What do I need to enter in the cross Product Calculator?
You enter Vector A and Vector B.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Exactly 3 components.
What is an example of a cross Product result?
With vector A 1, 2, 3 and vector B 4, 5, 6, the cross product is (-3, 6, -3).
Related calculators
- Dot Product Calculator — Dot product of two vectors.
- Solve for Unknown Fraction (Cross Multiplication) — Solve a/b = c/d when one value is unknown.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Euclidean Algorithm Calculator — GCD with every division step shown.
- Common Multiple Calculator — First common multiples of two numbers.
- Arithmetic Sequence Calculator — nth term and sum of an arithmetic sequence.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.