3D Cross Product of Two Vectors

Cross product u × v in 3D

The cross product takes two 3D vectors, u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), and gives back the vector u × v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁). What comes out is perpendicular to the plane that u and v span, and its length |u × v| = |u|·|v|·sin θ matches the area of the parallelogram the two vectors trace out. To find which way it points, use the right-hand rule: point your fingers from u toward v, and your thumb lands on u × v. For instance, (1, 0, 0) × (0, 1, 0) = (0, 0, 1), the familiar x̂ × ŷ = ẑ. A few properties worth remembering. It's anti-commutative, so u × v = -(v × u); parallel vectors give u × v = 0; and it distributes over addition. The dot product works in any dimension, but the cross product really lives only in 3D (7D is a strange exception).

Applications

It turns up all over mechanics: torque τ = r × F, angular momentum L = r × p, and the magnetic force on a moving charge F = qv × B (the Lorentz force). In 3D graphics you compute surface normals from two edge vectors to handle lighting. It also gives the area of a triangle as A = ½|u × v|, and a quick collinearity test for three points: the cross product of two edge vectors comes out zero.

FAQ

Why is the cross product a vector and the dot product a scalar? The cross product carries two pieces of information at once, a magnitude (the area) and a direction in space (the perpendicular). The dot product just measures how much one vector projects onto another, which boils down to a single number.

What does u × v = 0 mean? One of two things. Either one of the vectors is the zero vector, or u and v point along the same line so sin θ = 0. A vector crossed with itself, u × u, is always 0.

Does u × v exist in 2D? Not as a vector, no. What you get instead is a scalar, u₁v₂ - u₂v₁ (the z-component you'd see from the 3D version), and it tells you the signed area of the parallelogram.