Cross Product Calculator
Calculate the cross product of two 3D vectors and related properties.
Vector A
A = (1, 2, 3)
Vector B
B = (4, 5, 6)
Cross Product Formula
A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)
x-component:
(2)(6) - (3)(5) = -3
y-component:
(3)(4) - (1)(6) = 6
z-component:
(1)(5) - (2)(4) = -3
Cross Product A x B
(-3, 6, -3)
|A x B| = 7.3485
|A||A|
3.7417
|B||B|
8.7750
Angle
12.93
A . B
32.0000
Geometric Properties
Parallelogram Area7.3485
Triangle Area3.6742
Unit Vector of A x B
(-0.4082, 0.8165, -0.4082)
Properties of Cross Product
- The result is perpendicular to both input vectors
- A x B = -(B x A) (anti-commutative)
- A x A = 0 (zero vector)
- |A x B| = |A||B|sin(theta)
About Cross Product
Definition
The cross product of two vectors in 3D space is a vector that is perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors.
Applications
- Finding normal vectors to surfaces
- Calculating torque in physics
- Computing area of triangles/parallelograms
- 3D graphics and game development