Cross Product
The cross product of two 3D vectors produces a new vector perpendicular to both, with magnitude equal to the area of the parallelogram they form.
Formula
u \times v = (u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1)
Definition
The cross product takes two 3D vectors and produces a third vector perpendicular to both, with length equal to the area of the parallelogram formed by the original two. For $u = (u_1,u_2,u_3)$ and $v = (v_1,v_2,v_3)$: $u \times v = (u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1)$, computed via a $3 \times 3$ determinant with unit vectors $i, j, k$ in the first row; $|u \times v| = |u||v|\sin(\theta)$, with direction given by the right-hand rule, and unlike the dot product, the cross product is not commutative, $u \times v = -(v \times u)$. It is unique to 3D (and 7D via octonions), and is the Hodge dual of the wedge product, $u \times v = {*}(u \wedge v)$; the triple product $u \cdot (v \times w)$ equals the determinant of $[u,v,w]$ and gives the signed volume of the parallelepiped spanned by the three vectors.
Example
Point two fingers in different directions on the same flat surface: the cross product is a vector sticking straight up, perpendicular to both, the "normal" direction to the surface, essential in 3D graphics for lighting. By the right-hand rule, $i \times j = k$, $j \times k = i$, $k \times i = j$; for $u=(1,0,0)$, $v=(0,1,0)$, $u \times v = (0,0,1) = k$, pointing upward as expected. In electromagnetism, $F = q(v \times B)$ is the Lorentz force, torque is $\tau = r \times F$, and angular momentum is $L = r \times p$, making the cross product indispensable in classical and quantum mechanics.
Key Insight
The cross product gives you the "normal" direction to a flat surface defined by two vectors, and order matters, swapping the two vectors gives the opposite direction. In differential geometry, the cross product generalizes to the exterior product $\wedge$, which operates in any dimension and underpins differential forms, Stokes' theorem, and the calculus of multivariable integration.