Cross Product z-Component of Two 2D Vectors

Two vectors lying flat in the xy-plane have a cross product that points straight out of it, so only the z-component survives: a_xb_y − a_yb_x. It is the same expression as a 2×2 determinant, and it carries a sign that the magnitude form throws away. Positive means b lies counterclockwise from a; negative means clockwise; zero means the two are parallel. That sign is the workhorse of computational geometry: convex hull algorithms, point-in-triangle tests, polygon winding checks and line-segment intersection all reduce to asking which way three points turn.

Example: a = (3, 4), b = (1, 2) gives 3·2 − 4·1 = 2, a small positive number, so b sits slightly counterclockwise from a. Set the expression to zero and you get the parallelism test: with a = (2, 3) and b_y = 6, the vectors are parallel only when b_x = 4, since (2, 3) and (4, 6) are the same direction scaled. The usual mistake is dropping the minus sign or swapping the order: a × b = −(b × a), so reversing the arguments flips the answer's sign and the geometric conclusion with it.