Cross Product Calculator
A perpendicular vector, and the area it measures.
A perpendicular vector, and the area it measures.
Three components — the cross product exists only in 3D.
A × B
(0, 0, 1)
Magnitude 1 · 90° between them
The cross product returns a vector perpendicular to both inputs, with a length equal to the area of the parallelogram they span. That area is the fact that makes it useful in geometry rather than only in physics — halve it and you have the area of the triangle through the two vectors. It is anticommutative: A × B points exactly opposite to B × A. Which of the two you get follows the right-hand rule, and swapping the operands is the commonest source of a normal pointing into a surface instead of out of it. Parallel vectors give the zero vector, because they span no area at all. That is the test for collinearity, and it is why the result here says so plainly rather than returning three very small numbers.
How the Cross Product Calculator works
The cross product of two 3D vectors, with the unit normal and the parallelogram and triangle areas its magnitude represents. That area is what makes it useful in geometry rather than only in physics.
Also known as: vector perpendicular to two vectors · area of a triangle from vectors · normal vector calculator · right hand rule calculator
Where to go next
The Cross Product question rarely arrives on its own. These are the ones that usually come with it:
- Dot Product Calculator — The angle between two vectors, in any dimension.
- Inverse Matrix Calculator — With the condition number, so you know what it is worth.
- Eigenvalue Calculator — Complex eigenvalues named, not refused.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
What does the cross product give you?
A vector perpendicular to both inputs, whose length equals the area of the parallelogram they span. Direction follows the right-hand rule, and reversing the operands reverses it.
How do I find the area of a triangle from two vectors?
Half the magnitude of their cross product. The two vectors span a parallelogram; the triangle is exactly half of it.
Why is the cross product only defined in 3D?
Because only in three dimensions is there a unique direction perpendicular to two given vectors. In two dimensions there is no room, and in four or more there is a whole plane of perpendicular directions to choose from.
What does a zero cross product mean?
The vectors are parallel or antiparallel — they span no area at all. It is the standard test for collinearity of three points.
Why does the order matter?
The cross product is anticommutative: A × B points exactly opposite to B × A. Swapping the operands is the commonest cause of a surface normal pointing into an object instead of out of it.
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