Result vector
(0 ; 0 ; 1)
Components in the selected coordinate order.
Mathematics · Vectors & matrices
Cross product calculator — 3D vector and area
Calculate A×B, its norm and the areas of the associated parallelogram and triangle from two 3D vectors.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Vector norm
1
Euclidean length of the result vector.
Parallelogram area
1
Triangle area
0.5
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation details
01Rₓ
Formula
R₁ = A₂B₃ − A₃B₂Substitution
0 × 0 − 0 × 1Result
0
02Rᵧ
Formula
R₂ = A₃B₁ − A₁B₃Substitution
0 × 0 − 1 × 0Result
0
03R_z
Formula
R₃ = A₁B₂ − A₂B₁Substitution
1 × 1 − 0 × 0Result
1
04Result vector
Formula
R = A×BResult
(0 ; 0 ; 1)
05Vector norm
Formula
‖R‖ = √(∑ᵢ Rᵢ²)Substitution
√(0² + 0² + 1²)Result
1
06Parallelogram area
Formula
Sₚ = ‖A×B‖Substitution
1Result
1
07Triangle area
Formula
Sₜ = Sₚ/2Substitution
1 / 2Result
0.5
01Formulas and symbols
Formulas used
A×B
(AᵧB_z−A_zBᵧ, A_zBₓ−AₓB_z, AₓBᵧ−AᵧBₓ)
A×B: vector perpendicular to A and B, oriented by the right-hand rule.
Sₚ
Sₚ = ‖A×B‖
Sₜ = Sₚ/2
Sₚ: parallelogram area; Sₜ: triangle area.
| Symbol | Meaning |
|---|---|
| Aₓ | Vector A — ₓ |
| Aᵧ | Vector A — ᵧ |
| A_z | Vector A — 𝓏 |
| Bₓ | Vector B — ₓ |
| Bᵧ | Vector B — ᵧ |
| B_z | Vector B — 𝓏 |
| A×B | Result vector |
| ‖A×B‖ | Vector norm |
| Sₚ | Parallelogram area |
| Sₜ | Triangle area |
02Validation example
Reference numerical case
- A = (1, 0, 0), B = (0, 1, 0) give A×B = (0, 0, 1), norm 1, parallelogram area 1 and triangle area 0.5.
03How it works
In a right-handed frame, A×B is perpendicular to A and B and follows the right-hand rule. Its norm equals the parallelogram area; half that norm is the triangle area.
The parallelogram spanned by A and B has area equal to the cross-product norm. Its diagonal divides it into two triangles of equal area.
04Limits and conventions
Three real components per vector in a right-handed orthonormal frame. Parallel vectors give a zero cross product. The isometric projection is illustrative; exact components and areas are provided as text.
05Common mistakes
Order matters: B×A = −(A×B). The scalar dot product does not give the same direction or area.
06Practical questions
Why is the cross product perpendicular to both vectors?
The cross product A×B is defined as a vector normal to the plane containing A and B, with direction given by the right-hand rule. Its magnitude is ‖A‖‖B‖sinθ.
What does a zero cross product mean?
For nonzero vectors, A×B = 0 means the vectors are parallel or antiparallel. It also occurs if either input vector is the zero vector.
07Technical references
- Calculus Volume 3 — the cross product — OpenStax