Cross Product Calculator: Vector Multiplication
Calculate the cross product of two 3D vectors.
Cross Product Calculator
Calculate cross product of two 3D vectors
Quick Tips
- • Result is perpendicular to both input vectors
- • Magnitude equals area of parallelogram formed by vectors
- • A × B = - (B × A) (anti-commutative)
- • For parallel vectors, cross product is zero vector
What is a Cross Product?
The cross product of two 3D vectors A and B gives a new vector A × B that is perpendicular to both. The direction is determined by the right-hand rule.
It’s used in physics and engineering for torque, rotation, surface normals, and area calculations.
Cross Product Formula
If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), then:
A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
Determinant Method (How It’s Built)
| Component | Expression | Meaning |
|---|---|---|
| x | a₂b₃ − a₃b₂ | Uses y and z parts |
| y | a₃b₁ − a₁b₃ | Uses z and x parts |
| z | a₁b₂ − a₂b₁ | Uses x and y parts |
Step-by-Step Calculation
- 1Write A = (a1, a2, a3) and B = (b1, b2, b3).
- 2Compute x = a2*b3 − a3*b2.
- 3Compute y = a3*b1 − a1*b3.
- 4Compute z = a1*b2 − a2*b1.
- 5Combine the result as (x, y, z).
Worked Examples
| A | B | A × B |
|---|---|---|
| (1, 2, 3) | (4, 5, 6) | (-3, 6, -3) |
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) |
| (0, 1, 0) | (1, 0, 0) | (0, 0, -1) |
| (2, -1, 3) | (0, 4, 1) | (-13, -2, 8) |
Important: Cross product is not commutative. That means A × B = −(B × A). If you swap vectors, the direction flips.
Magnitude Meaning (Area)
The magnitude of the cross product is:
|A × B| = |A| |B| sin(θ)
This equals the area of the parallelogram formed by A and B. Half of that is the area of the triangle formed by the two vectors.
Key Properties (Quick Facts)
- A × B is perpendicular to both A and B.
- A × A = (0, 0, 0).
- If A and B are parallel, A × B = (0, 0, 0).
- A × B = −(B × A).
- Scaling: (kA) × B = k(A × B).
Common Uses of Cross Product
- Torque: τ = r × F
- Angular momentum: L = r × p
- Surface normal vector in 3D graphics (lighting/shading)
- Area of parallelogram/triangle from two vectors
- Finding perpendicular direction in 3D geometry
Visual: A, B and A × B (Right-Hand Rule)
Cross Product FAQs
What is the cross product of two vectors?
It’s a vector perpendicular to both input vectors. Its direction follows the right-hand rule and its magnitude represents area.
Is cross product defined in 2D?
Not in the same way. In 2D, people often return a scalar representing the z-component you'd get if you treat vectors as 3D with z = 0.
What happens if the vectors are parallel?
If vectors are parallel (θ = 0° or 180°), sin(θ) = 0, so A × B = (0, 0, 0).
Why does swapping vectors change the sign?
Because A × B and B × A point in opposite directions. So A × B = −(B × A).
What does |A × B| mean?
It equals |A||B|sin(θ), which is the area of the parallelogram formed by A and B.
How is cross product used in physics?
Common examples include torque (r × F) and angular momentum (r × p).
What is a normal vector in 3D graphics?
A normal vector is perpendicular to a surface. Cross product is used to compute it from two edges of a polygon.
Can the result be the zero vector even if inputs are non-zero?
Yes. If A and B are parallel or one of them is the zero vector, the cross product is zero.