What does the determinant-like thing look like when you take the cross product of a vector by itself?
Rows two and three will be the same.
Recall from the dark ages of high school math that if two rows of a
determinant are the same, then it evaluates to zero.
Each co-factor evaluates to zero, resulting in
The same will happen if one row is a multiple of another.
These results reflect what we have already seen
with geometrical vectors:
The other properties of the cross product of geometrical vectors are also true of the cross product of their column matrix representations.
What were those properties again?