Cross Product Calculator
Compute the 3D cross product (vector product) A × B. Shows the perpendicular vector, parallelogram area, triangle area, and angle. Includes right-hand rule explanation.
Vector A (3D)
Vector B (3D)
| Property | Formula | Result |
|---|---|---|
| Cross product A × B | (ay·bz−az·by, az·bx−ax·bz, ax·by−ay·bx) | Vector ⊥ to A and B |
| Magnitude |A × B| | |A||B|sin(θ) | Scalar = parallelogram area |
| Anti-commutativity | A × B = −(B × A) | Swap order → flip direction |
| Self cross product | A × A = 0 | Zero vector always |
| Parallel vectors | A × B = 0 if A ∥ B | sin(0°) = 0 |
Cross Product Applications
- Torque in Physics: τ = r × F. The torque about a pivot equals position vector crossed with force. Direction tells you clockwise vs counter-clockwise rotation.
- 3D Graphics — Surface Normals: Given two edges of a triangle (vectors), their cross product is the normal to the surface — essential for lighting calculations in every 3D game and rendering engine.
- Magnetic Force: F = q(v × B). The force on a moving charge in a magnetic field is the cross product of velocity and magnetic field.
- Area Calculation: The magnitude |A × B| equals the area of the parallelogram with sides A and B. Half of this is the triangle area — useful in computational geometry.
Frequently Asked Questions
What's the difference between dot product and cross product?
The dot product (A · B) produces a scalar and measures how parallel two vectors are. The cross product (A × B) produces a vector perpendicular to both and measures how perpendicular two vectors are. Both use the angle θ between vectors: dot uses cos(θ), cross uses sin(θ). Dot product works in any dimension; cross product only in 3D.
How does the determinant formula relate to the cross product?
The cross product can be computed as a formal determinant: A × B = det([[î, ĵ, k̂], [ax, ay, az], [bx, by, bz]]), expanding along the first row. This is how it appears in most linear algebra textbooks (UK A-Level Further Maths, IB HL, US university curricula).