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: