Calculates the cross product of two three-dimensional vectors, producing a new vector perpendicular to both.
How it works
Each component of the result combines the other two components of both input vectors using the standard cross-product determinant pattern.
What this does not include
This does not include the dot product, which produces a single number rather than a vector — for that, use this site’s dot product calculator instead.
How to use this calculator
- Enter both vectors’ three components.
A worked example
Vectors (1,0,0) and (0,1,0): cross product = (0, 0, 1) — perpendicular to both input vectors, pointing along the z-axis.
Vectors (2,3,4) and (5,6,7): cross product = (−3, 6, −3).
What the variables mean
| Variable | Meaning |
|---|---|
| Vector A (a1, a2, a3) | First 3D vector’s components |
| Vector B (b1, b2, b3) | Second 3D vector’s components |
Edge cases worth knowing
Unlike the dot product, the cross product returns another vector, not a single number — one that’s perpendicular to both original vectors, as the first example shows directly.
Cross product order matters — A × B and B × A point in opposite directions (same magnitude, flipped sign), unlike the dot product, which is the same either way.
Frequently asked questions
Why is the result a vector instead of a number?
The cross product specifically measures a new direction perpendicular to both input vectors, along with a magnitude related to the area of the parallelogram they form.
Does the order of the vectors matter?
Yes — reversing the order flips the sign of every component in the result, since a × b = −(b × a).
What’s a real-world use for the cross product?
Finding a surface’s normal vector in 3D graphics, or computing torque and angular momentum in physics, both rely on the cross product.
