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

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