Cross Product Calculator Formula

Understand the math behind the cross product calculator. Each variable explained with a worked example.

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Formulas Used

Cx

cx = a2*b3 - a3*b2

Cy

cy = a3*b1 - a1*b3

Cz

cz = a1*b2 - a2*b1

Magnitude

magnitude = sqrt(pow(a2*b3 - a3*b2, 2) + pow(a3*b1 - a1*b3, 2) + pow(a1*b2 - a2*b1, 2))

Variables

VariableDescriptionDefault
a1Vector A: x2
a2Vector A: y3
a3Vector A: z4
b1Vector B: x5
b2Vector B: y6
b3Vector B: z7

How It Works

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

    • Worked Example

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

    a1 = 2a2 = 3a3 = 4b1 = 5b2 = 6b3 = 7
    1. 01x: 3×7 - 4×6 = 21 - 24 = -3
    2. 02y: 4×5 - 2×7 = 20 - 14 = 6
    3. 03z: 2×6 - 3×5 = 12 - 15 = -3
    4. 04A × B = (-3, 6, -3)
    5. 05|A × B| = √(9+36+9) = √54 ≈ 7.3485

    Frequently Asked Questions

    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.

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