Cross Product

The cross product and triple product calculator finds the cross product a × b with the determinant expansion, its length (the area of a parallelogram and a triangle), the scalar triple product (a, b, c) — the volume of a parallelepiped and a tetrahedron — and the area of a triangle from the coordinates of three points in the plane or in space.

Cross product

a × b = (y₁z₂ − z₁y₂, z₁x₂ − x₁z₂, x₁y₂ − y₁x₂)

It is convenient to compute it as a determinant with the unit vectors i, j, k in the first row. The vector a × b is perpendicular to both vectors, and its length |a × b| = |a| · |b| · sin φ equals the area of the parallelogram built on a and b.

Example 1. a = (1, 2, 3), b = (4, 5, 6).
a × b = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3),   |a × b| = √54 = 3√6.

Scalar triple product

(a, b, c) = (a × b) · c is the determinant of the coordinates of the three vectors. Its absolute value is the volume of the parallelepiped on the vectors; the tetrahedron volume is 6 times smaller. If (a, b, c) = 0, the vectors are coplanar.

Example 2. a = (1, 2, 3), b = (0, 1, 4), c = (5, 6, 0).
(a, b, c) = 1 · (0 − 24) − 2 · (0 − 20) + 3 · (0 − 5) = 1, V_tetrahedron = 1/6.

Area of a triangle from coordinates

S = ½ · |AB × AC|,   in the plane: S = ½ · |(x_B − x_A)(y_C − y_A) − (y_B − y_A)(x_C − x_A)|