The vector cross product can be represented by a third-order determinant,

Then expand according to the first line,

There should be a negative sign in front of the second-order determinant in the middle.

It is required by the algebraic cofactor (i.e.-1+2 power).

The absence of negative signs before and after is also the result of algebraic remainder.

(One is 1 power of-1 and the other is 1+3 power of-1).