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Cross Product Calculator

Calculate the cross product of two 3D vectors, plus the dot product, magnitudes and the angle between them. Free vector calculator.

a × b
(-3, 6, -3)
|a × b| (parallelogram area)
7.348469
Triangle area
3.674235
Dot product a · b
32
|a|
3.741657
|b|
8.774964
Angle between a and b
12.9332°

Cross product of two vectors

The cross product a × b of two 3D vectors is a new vector at right angles to both. Its length equals the area of the parallelogram they span. a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For a = (1, 2, 3) and b = (4, 5, 6), a × b = (−3, 6, −3). The dot product a · b = 32 measures how much they point the same way, and gives the angle between them: 12.93°.

How to use it

  1. 1

    Enter vector a

    Its x, y and z components.

  2. 2

    Enter vector b

    Its x, y and z components.

  3. 3

    Read the results

    Cross product, dot product, magnitudes and angle.

Frequently asked questions

Is the cross product commutative?

No. b × a = −(a × b): it points the opposite way.

What does a zero cross product mean?

The vectors are parallel (or one is zero).

How do I find the angle between vectors?

cos θ = (a · b) ÷ (|a| |b|).

Is anything stored?

No. The calculation runs in your browser.

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