Areas from the cross product

|a × b| = |a||b| sin θ is base × height for the parallelogram with adjacent sides a and b, so that parallelogram has area |a × b|.

The triangle with sides a and b from one corner is half of it: area ½|a × b|. For corners A, B, C use ½|AB × AC|.

If d₁ and d₂ are the diagonals of a parallelogram, its area is ½|d₁ × d₂|.

Worked: A(1, 1, 1), B(1, 2, 3), C(2, 3, 1) give AB = ĵ + 2k̂, AC = î + 2ĵ, AB × AC = −4î + 2ĵ − k̂, so the triangle's area is ½√21.

Worked: sides 3î + ĵ + 4k̂ and î − ĵ + k̂ give a × b = 5î + ĵ − 4k̂, so the parallelogram's area is √42.

Three points are collinear exactly when AB × AC = 0 (the triangle has no area). For A(1, 1, 1), B(2, 5, 0), C(3, 2, −3), D(1, −6, −1), CD = −2AB, so AB and CD are collinear and the angle between them is π.