What is the cross product?

The cross product of two 3D vectors produces a vector perpendicular to both inputs. Its magnitude equals the area of the parallelogram formed by the two vectors.

Does the cross product work in 2D?

The cross product is only defined for 3D vectors. In 2D, you can treat vectors as 3D with z=0, and the result will point entirely in the z direction.

What does the magnitude represent?

The magnitude of A × B equals |A|×|B|×sin(theta), which is the area of the parallelogram spanned by the two vectors.

Cross Product Calculator

Calculate the cross product of two 3D vectors. Shows the resulting vector and its magnitude.

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Vector A: x

Vector A: y

Vector A: z

Vector B: x

Vector B: y

Vector B: z

-3.0000

Magnitude7.3485

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Cross Product

Formula

A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

Properties

The result is perpendicular to both input vectors

A × B=A×B × sin(theta) (area of parallelogram)

A × B = -(B × A) (anticommutative)

If A × B = (0,0,0), the vectors are parallel

Example Calculation

Cross product of (2,3,4) and (5,6,7).

01x: 3×7 - 4×6 = 21 - 24 = -3
02y: 4×5 - 2×7 = 20 - 14 = 6
03z: 2×6 - 3×5 = 12 - 15 = -3
04A × B = (-3, 6, -3)
05|A × B| = √(9+36+9) = √54 ≈ 7.3485

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Cross Product Calculator

Cross Product

Formula

Properties

Example Calculation

Frequently Asked Questions

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