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
**i** + 0**j** + 0**k** = **0**.

The same will happen if one row is a multiple of another.
These results reflect what we have already seen
with geometrical vectors:
**u** **×** **u** = **0**.

The other properties of the cross product of geometrical vectors are also true of the cross product of their column matrix representations.