Solutions, Homework 2, Math 152-205, additional problem:

(Part a)

The sign changes in operations (3),(4),(5) when you exchange  a  and  b ;
Operations (1),(2) remain the same when you exchange a  and  b.

(Part b)

The dot product and the cosine appear first and together in the notes (in
Section 2.3, see the table of contents), and are closely related (the dot
product is the cosine times the lengths of the vectors).  These are operations
(1),(2), that don't change sign.

The cross product, determinant, and sine appear together later in the notes (in
Section 2.4), and are closely related (the cross product is defined by a  3 x 3
determinant, and the norm of a cross product or size of a 2 x 2 determinant
related to the sine times the lengths of the vectors).  These are operations
(3),(4),(5), that do change signs.