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Cross Product Calculator

Find the cross product of two vectors.

How it is calculated

Enter the three components of vector a and of vector b. All six fields are required, and the vectors are unitless: the answer carries whatever units the components imply. The cross product c = a × b is built component by component:

cx = aybz − azby

cy = azbx − axbz

cz = axby − aybx

The tool also gives the magnitude |c| = √(cx² + cy² + cz²), which equals the area of the parallelogram spanned by a and b. Everything is rounded to two decimals. Parallel vectors give a zero vector. The order matters: b × a = −(a × b).

Worked example

Let a = (3, −3, 1) and b = (4, 9, 2).

  1. x: (−3)(2) − (1)(9) = −6 − 9 = -15.00.
  2. y: (1)(4) − (3)(2) = 4 − 6 = -2.00.
  3. z: (3)(9) − (−3)(4) = 27 + 12 = 39.00.
  4. Magnitude: √(225 + 4 + 1521) = √1750 = 41.83.

So a × b = (-15.00, -2.00, 39.00), and the parallelogram formed by the two vectors has an area of 41.83 square units.

Try this example →

Common mistakes

  • Swapping the vectors. a × b and b × a point in opposite directions, so the x, y and z values change sign. The magnitude stays the same.
  • Dropping a minus sign in the middle term. The y component is azbx − axbz, which is easy to write the wrong way round when doing it by hand.
  • Using 2D vectors without a third component. The tool needs all six numbers; for planar vectors enter 0 for z, and the result has only a z component.
  • Expecting a single number like the dot product. The cross product is a vector; only the Magnitude line is a scalar.
  • Reading a zero result as an error. A zero vector is correct when a and b are parallel, or when either one is the zero vector.

About this calculator

This calculator computes the cross product of two 3D vectors, a × b, which — unlike the dot product — returns a new vector perpendicular to both inputs, with a magnitude equal to |a||b|sin(θ) and a direction given by the right-hand rule.

It's used in 3D graphics and game development to compute surface normals (the direction a face is pointing, essential for lighting), in physics for torque (τ = r × F) and angular momentum calculations, and in engineering for finding a plane's orientation from two vectors lying in it. The calculator also reports the magnitude of the resulting vector, which conveniently equals the area of the parallelogram formed by the two input vectors.

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Comments (1)

  • Editorial team

    A quick way to check your result is to take its dot product with each input vector: both must come out as zero, because the cross product is perpendicular to both. Half the magnitude is also the area of the triangle formed by the two vectors, so a = (2, 0, 0) and b = (0, 3, 0) give a magnitude of 6 and a triangle area of 3. Where do you use it, such as surface normals, torque or something else?

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