Parallelogram Area from Two Vectors

Two vectors sharing a corner sweep out a parallelogram, and its area is the absolute value of the cross product's z-component: |a_xb_y − a_yb_x|. The result needs no angle, no perpendicular height and no trigonometry, just four coordinates. Sides a = (3, 1) m and b = (1, 4) m enclose |12 − 1| = 11 m². The construction is a special case of Hermann Grassmann's exterior product from his 1844 Ausdehnungslehre, a book so far ahead of its style that almost nobody read it; Grassmann gave up mathematics and became a distinguished Sanskrit scholar instead, and the algebra he invented was only rediscovered decades later.

Infinitely many pairs of vectors enclose the same area, so the area alone recovers nothing; with the other three components held, one component comes back, but twice, once for each sign of the determinant. The solver returns the one with b counterclockwise from a. The determinant's sign, stripped away by the absolute value, is worth keeping when you compute it by hand: it tells you whether b lies clockwise or counterclockwise from a, and polygon area algorithms depend on exactly that.