Cross Product Calculator
Find the cross product of two 3D vectors with magnitude and direction.
Inputs
A × B
[-3, 6, -3]
||A × B||
7.348469
Parallelogram Area
7.348469
Step by step
A
= [1, 2, 3]
B
= [4, 5, 6]
A × B = [A₂B₃−A₃B₂, A₃B₁−A₁B₃, A₁B₂−A₂B₁]
= [-3, 6, -3]
||A × B|| (area of parallelogram)
= 7.348469
How it works
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
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.