Definition: Cross product on
Take some . Cross product of the vector will be based on the determinant of the components of in the following way:
- The cross product will be a vector also in
that is orthogonal to both of the vectors - We can see this by taking the plane formed by
and , and creating a vector that is orthogonal to that plane
- We can see this by taking the plane formed by
Proof: Cross product orthogonality For our first case, we have to work to show that
. When expanded, this simplifies to zero. For our second case, we have to show that Which when expanded, also simplifies to zero.
Proof: Say that
. Then, TBA