3D Cross Product

Cross product u × v in 3D

The cross product of u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃) comes out as u × v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁). The result sits perpendicular to the plane that u and v span, and its length |u × v| = |u|·|v|·sin θ matches the area of the parallelogram they form. Which way does it point? Use the right-hand rule: curl your fingers from u toward v and the thumb lands on u × v. For instance, (1, 0, 0) × (0, 1, 0) = (0, 0, 1), the familiar x̂ × ŷ = ẑ. The operation is anti-commutative, and it drops to zero whenever u and v run parallel.

Applications

Mechanics leans on it for torque τ = r × F and angular momentum L = r × p, and electromagnetism uses it for the Lorentz force on a moving charge F = qv × B. In 3D computer graphics it builds surface normals from two edge vectors so lighting and shading work. It also gives the triangle area A = ½|u × v| and lets you test whether three points are collinear.

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 that is the area and an orientation in space along the perpendicular direction. The dot product only measures projection, so a single number is all it needs.

What does u × v = 0 mean? One of two things: either one vector is zero, or u and v are parallel so sin θ = 0. A vector crossed with itself, u × u, is always 0 for the same reason.

Does u × v exist in 2D? Not as a vector, no. What you get in 2D is the scalar u₁v₂ - u₂v₁, which is the signed area of the parallelogram.