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Parallelogram Area from Two Vectors

A=∣axby−aybx∣A = \left|a_x b_y - a_y b_x\right|

Worked example: a = (3, 1) m, b = (1, 4) m → 11 m² — press Try an example to run it live, then adjust anything.

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Parallelogram Area from Two Vectors explained

(ax, ay)(bx, by)A

Two vectors sharing a corner sweep out a parallelogram, and its area is the absolute value of the cross product's z-component: |axby − aybx|. 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.

The solver handles this one for the area only. That is not a limitation of the algebra but of the geometry: infinitely many pairs of vectors enclose the same area, so a single component cannot be recovered from it. 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.

Parallelogram Area from Two Vectors formula

A=∣axby−aybx∣A = \left|a_x b_y - a_y b_x\right|
Where
  • AA= Parallelogram area (m²)
  • axa_x= x-component of a (m)
  • aya_y= y-component of a (m)
  • bxb_x= x-component of b (m)
  • byb_y= y-component of b (m)

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