The cross product formula
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
It comes from expanding the determinant with unit vectors i, j and k in the top row:
| i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |
The right-hand rule
Point the fingers of your right hand along a, curl them toward b, and your thumb points in the direction of a × b. That is why i × j = k but j × i = −k.
Where it is used
- Physics: torque (r × F), angular momentum and the magnetic force on a moving charge.
- 3D graphics: finding the normal vector of a surface for lighting.
- Geometry: area of a triangle in space is half of |a × b|.
Working with larger arrays of numbers? See the matrix calculator.
Last reviewed October 2026.
Frequently asked questions
What is the cross product?
The cross product of two 3D vectors a and b is a new vector that is perpendicular to both. Its length equals the area of the parallelogram the two vectors span, and its direction follows the right-hand rule.
How is the cross product different from the dot product?
The cross product gives a vector; the dot product gives a single number (a scalar). The dot product measures how much two vectors point the same way, while the cross product measures how perpendicular they are.
What does a zero cross product mean?
If a × b = (0, 0, 0), the vectors are parallel (or one of them is zero). There is no unique perpendicular direction and the parallelogram has no area.
Is the cross product commutative?
No. Swapping the order flips the direction: b × a = −(a × b). The length stays the same.
Can I find the cross product of 2D vectors?
Enter 0 for the z components. The result points straight along the z-axis, and its z value is the signed area of the parallelogram in the plane.