Cross Product
Calculus & Advanced MathThe 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 that is perpendicular to both of them. The length of this new vector equals the area of the parallelogram formed by the original two.
Example
Point two fingers in different directions on the same flat surface. The cross product is a vector sticking straight up (out of the surface), perpendicular to both fingers.
Key Insight
The cross product gives you the "normal" direction to a flat surface defined by two vectors, which is essential in 3D graphics for lighting calculations.
Definition
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 the determinant of a $3 \times 3$ matrix with unit vectors $i, j, k$ in the first row. $|u \times v| = |u||v|\sin(\theta)$, direction by right-hand rule.
Example
$i \times j = k$, $j \times k = i$, $k \times i = j$ (right-hand rule). For $u=(1,0,0)$, $v=(0,1,0)$: $u \times v = (0 \times 0-0 \times 1, 0 \times 0-1 \times 0, 1 \times 1-0 \times 0) = (0,0,1) = k$. Points upward as expected.
Key Insight
Unlike the dot product, the cross product is NOT commutative: $u \times v = -(v \times u)$. Order matters, and swapping gives the opposite direction.
Definition
The cross product is unique to 3D (and 7D via octonions). It 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 the $3 \times 3$ matrix $[u,v,w]$ and gives the signed volume of the parallelepiped spanned by the three vectors.
Example
In electromagnetism: $F = q(v \times B)$ (Lorentz force). Torque: $\tau = r \times F$. Angular momentum: $L = r \times p$. The cross product is indispensable in classical and quantum mechanics.
Key Insight
In differential geometry, the cross product generalizes to the exterior product ∧, which operates in any dimension and underpins differential forms, Stokes' theorem, and the calculus of multivariable integration.