Vector Product

Right-handed and left-handed systems. If you align the fingers of your right hand along $\vec a$ and curl them towards $\vec b$ (through the angle less than $180^\circ$), your thumb points along $\vec a\times\vec b$; curling the other way (from $\vec b$ towards $\vec a$) gives $\vec b\times\vec a$, pointing in the opposite direction. A Cartesian system is right-handed if $\hat i,\hat j,\hat k$, taken along the positive axes in that order, follow this same right-hand rule; it is left-handed if the sense of $\hat k$ is reversed. We use right-handed systems throughout.

Definition. For non-zero vectors $\vec a,\vec b$ with included angle $\theta$ ($0\le\theta\le\pi$), the vector product (or cross product) is $\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat n,$ where $\hat n$ is the unit vector perpendicular to both $\vec a$ and $\vec b$, chosen so that $\vec a,\vec b,\hat n$ form a right-handed system (in that order). Since the result is itself a vector, this is the vector product; the symbol $\times$ gives it the name cross product.

Direction matters — order matters. Because $\hat n$ is fixed by the order $\vec a,\vec b$, swapping the order reverses the direction of $\hat n$, and hence of the whole product. …