The cross product (or vector product) between two vectors produces a vector that is orthogonal to both input vectors, i.e., for some vector , :

We can also relate the angle between two vectors to the cross product, where .

The key implication of this is that the magnitude of the cross product is the area of the parallelogram created by and .

Determinants are also very useful tools for computing the cross product by hand in . We can do Laplace expansion where the first row are the unit vectors , and the corresponding next two rows are the two vectors. In MATLAB, we can use cross(v, w).

Properties

The cross product is non-commutative:

If and only if is parallel to :

In general, the cross product fails to be associative:

See also