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Cross Product Calculator

Enter two 3D vectors to calculate A × B and the magnitude of the perpendicular result.

Enter vector components

Your data stays in this browser.

Vector A
Vector B

Result

Enter the components, then select Calculate.

How to use the cross product calculator

This page calculates the ordered 3D cross product A × B. The order is visible throughout the form and result because reversing it changes the direction.

  1. Enter x, y, and z for vector A.
  2. Enter x, y, and z for vector B.
  3. Select Calculate to compute A × B.
  4. Read the three result components and the magnitude |A × B|.
  5. Copy the result when needed. Clear beside A or B removes only that vector’s components and invalidates the old result.

Cross product formula and direction

For A = (a₁,a₂,a₃) and B = (b₁,b₂,b₃), A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The answer is a 3D vector perpendicular to both inputs whenever the result is nonzero.

The operation is not commutative: B × A = −(A × B). This calculator always follows the displayed order A × B. The right-hand rule describes the corresponding orientation, but no interactive 3D scene is needed to perform the component calculation.

Magnitude and the zero result

The magnitude is |A × B| = √(x²+y²+z²) for the resulting vector. Geometrically it also equals |A||B|sin θ, the area of the parallelogram spanned by A and B.

Parallel or opposite-direction vectors produce the zero vector because sin 0° and sin 180° are zero. A zero input vector also produces zero. The calculator reports this as a normal result rather than an input error; the result alone cannot distinguish which of those cases caused it.

Uses, limits, and common mistakes

  • Find a normal vector for a plane defined by two nonparallel direction vectors.
  • Check orientation in geometry, graphics, and engineering calculations while preserving A × B order.
  • Use the magnitude as the parallelogram area when the vector components share compatible length units.
  • This page supports 3D Cartesian components only. It does not add a 2D scalar extension, matrices, torque units, or a 3D editor.
  • Enter one finite number per field. Display values are rounded to at most two decimals while internal arithmetic keeps normal floating-point precision.

Worked examples

A = (1,2,3), B = (4,5,6): A × B = (−3,6,−3), with magnitude √54 ≈ 7.35.

Reversing those vectors gives B × A = (3,−6,3), exactly the negative of the first result.

A = (1,2,3), B = (2,4,6): the vectors are parallel and A × B = (0,0,0), magnitude 0.

Frequently asked questions

Why does order matter?

The cross product is anti-commutative: reversing the inputs reverses every result component.

Is a zero cross product an error?

No. It is the correct result for parallel vectors or when either input vector is zero.

What does the magnitude represent?

It is the length of the perpendicular result vector and the area of the parallelogram spanned by the inputs.

Can I use 2D vectors?

Not on this page. Its scope is the standard 3D vector cross product.

Does the result prove perpendicularity?

A nonzero cross product is perpendicular to both inputs. The zero vector has no unique direction.