The cross product is an operation that finds a third vector, , that is orthogonal to two vectors, and . The problem of finding a vector orthogonal to another two comes up frequently in math and physics, which is the motivation for the cross product.
Suppose we have two vectors and that are in . We can conclude that is orthogonal to both vectors if:
If we were to solve this linear system, we would end up with the following vector for .
This tells us that in , the cross product operation can be defined as:
For now, we will only worry about cross products in . It is possible to do a cross product in , however it is much more complex and out of the scope of introduction level Linear Algebra.
There are a set of properties that apply to the cross product operation. Let , and . The following properties hold:
One example of a cross product application is using it to find the normal of a plane. A plane has an equation in the form , where and are linearly independent. A normal vector is vector that is perpendicular to both and . Therefore, we can conclude that