The Cross Product (Vector Product)
A powerful operation that takes two vectors in three-dimensional space and produces a new vector perpendicular to both.
1. Definition in Space
Given two vectors $\mathbf{a}$ and $\mathbf{b}$ in three-dimensional space, their cross product (also called the vector product), written $\mathbf{a} \times \mathbf{b}$, is a vector defined by three properties:
where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$ (with $0 \le \theta \le \pi$).
$\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, and therefore perpendicular to the plane that contains them.
The sense (up or down from the plane) is fixed by the right-hand rule.
The Right-Hand Rule & the Screwdriver Analogy
Point the fingers of your right hand along $\mathbf{a}$, then curl them toward $\mathbf{b}$ through the smaller angle. Your thumb points in the direction of $\mathbf{a} \times \mathbf{b}$.
Screwdriver analogy: Imagine a right-handed (standard) screwdriver. Align the handle with $\mathbf{a}$ and turn the tip toward $\mathbf{b}$. The direction the screw advances is the direction of $\mathbf{a} \times \mathbf{b}$.