The cross product gives the axis of rotation for torque (τ = r × F), the normal vector of a plane, the area of a triangle in 3D and the direction of magnetic forces. It follows the right-hand rule.
How to use the Cross Product Calculator
- a – x
- a – y
- a – z
- b – x
- b – y
- b – z
Results update as you type, and everything is calculated in your browser.
How it works
Each component is a 2 × 2 determinant of the other two coordinates. The magnitude equals the area of the parallelogram spanned by a and b; half of it is the triangle area.
Inputs explained
| Field | What to enter |
|---|---|
| a – x | A number |
| a – y | A number |
| a – z | A number |
| b – x | A number |
| b – y | A number |
| b – z | A number |
What each result means
- c = a × b: x = [a – y]×[b – z] – [a – z]×[b – y]
- c = a × b: y = [a – z]×[b – x] – [a – x]×[b – z]
- c = a × b: z = [a – x]×[b – y] – [a – y]×[b – x]
- |a × b| (parallelogram area) = sqrt(([a – y]×[b – z] – [a – z]×[b – y])^2 + ([a – z]×[b – x] – [a – x]×[b – z])^2 + ([a – x]×[b – y] – [a – y]×[b – x])^2)
- Triangle area = sqrt(([a – y]×[b – z] – [a – z]×[b – y])^2 + ([a – z]×[b – x] – [a – x]×[b – z])^2 + ([a – x]×[b – y] – [a – y]×[b – x])^2)÷2
Example values
How the main result changes when a – x changes and everything else stays at the example values above:
| a – x | c = a × b: x |
|---|---|
| 1 | -3 |
| 1.5 | -3 |
| 2 | -3 |
| 2.5 | -3 |
| 3 | -3 |
| 4 | -3 |
Formula
a × b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x)
|a × b| = |a||b| sin θ
Worked example
Inputs — a – x: 2; a – y: 3; a – z: 4; b – x: 5; b – y: 6; b – z: 7.
Results — c = a × b: x = -3; c = a × b: y = 6; c = a × b: z = -3; |a × b| (parallelogram area) = 7.34846923; Triangle area = 3.67423461; Angle between a and b = 7.4758 °.
Key terms
- Cross product
- A vector perpendicular to two given 3D vectors.
Frequently asked questions
Is a × b the same as b × a?
No: b × a = −(a × b).
What does a zero cross product mean?
The vectors are parallel (or one is zero).
Does the cross product exist in 2D?
Not as a vector; the scalar a_x b_y − a_y b_x gives the signed area.
What is the right-hand rule?
Point your fingers along a, curl them towards b, and your thumb points along a × b.