Cross Product Calculator
The three-dimensional cross product returns a vector perpendicular to the two input vectors. Its direction follows the right-hand rule, and its magnitude equals the area of the parallelogram spanned by the vectors.
How to use this calculator
- Enter x, y and z components for both vectors. The result is returned as an ordered three-component vector and its magnitude.
- Check the labels and units before applying the result to a real task.
- Change one input at a time when comparing scenarios.
Formula and method
A×B = (aᵧb_z−a_zbᵧ, a_zbₓ−aₓb_z, aₓbᵧ−aᵧbₓ)
- A×B = (aᵧb_z−a_zbᵧ, a_zbₓ−aₓb_z, aₓbᵧ−aᵧbₓ)
- The vector direction depends on coordinate orientation and order. Swapping the input vectors reverses the sign of the result.
Worked examples
Worked example
For A = (1, 2, 3) and B = (4, 5, 6), calculate the cross product.
A×B = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (−3, 6, −3).
A×B = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (−3, 6, −3).
Understanding the result
The three-dimensional cross product returns a vector perpendicular to the two input vectors. Its direction follows the right-hand rule, and its magnitude equals the area of the parallelogram spanned by the vectors.
The vector direction depends on coordinate orientation and order. Swapping the input vectors reverses the sign of the result.
Important notes
The vector direction depends on coordinate orientation and order. Swapping the input vectors reverses the sign of the result.
Displayed values are rounded for readability; retain appropriate precision when continuing a calculation.
Frequently asked questions
Why does switching A and B change the answer?
The cross product is anti-commutative: B×A = −(A×B). The magnitude is unchanged, but the direction reverses.
What is the cross-product magnitude used for?
Its magnitude gives the area of the parallelogram defined by the two vectors, and half that value is the corresponding triangle area.
What if the vectors are parallel?
Parallel or anti-parallel vectors have a zero cross product because the spanned parallelogram has zero area.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- Area — Understand area as the amount of two-dimensional surface inside a shape and review common formulas.
This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.