Cross Product Calculator
Find the cross product of two 3D vectors with magnitude and direction.
About Cross Product Calculator
The cross product A × B of two 3D vectors produces a vector perpendicular to both A and B. Its magnitude equals the area of the parallelogram spanned by A and B. The direction follows the right-hand rule. The cross product is only defined in 3 dimensions (and 7 dimensions).
Formulas
Cross Product
A × B = [A₂B₃−A₃B₂, A₃B₁−A₁B₃, A₁B₂−A₂B₁]
- A
- First 3D vector
- B
- Second 3D vector
Magnitude
||A × B|| = ||A|| · ||B|| · sin(θ)
- θ
- Angle between A and B
Worked example
With the default values below, Cross Product Calculator returns A × B of [-3, 6, -3]. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Vector A (comma or space separated)
- 1, 2, 3
- Vector B (comma or space separated)
- 4, 5, 6
Results
- A × B
- [-3, 6, -3]
- ||A × B||
- 7.348469
- Parallelogram Area
- 7.348469
Step by step
- 1A= [1, 2, 3]
- 2B= [4, 5, 6]
- 3A × B = [A₂B₃−A₃B₂, A₃B₁−A₁B₃, A₁B₂−A₂B₁]= [-3, 6, -3]
- 4||A × B|| (area of parallelogram)= 7.348469
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Is the cross product commutative?
No. A × B = −(B × A). The cross product is anti-commutative — swapping the operands reverses the direction.
When is the cross product zero?
When the vectors are parallel (or one is the zero vector). Parallel vectors span no area, so the cross product magnitude is zero.