$$\require{cancel}$$
In analogy with the definition of torque, $$\boldsymbol{\tau}=\boldsymbol{r} \times \boldsymbol{F}$$ as the rotational counterpart of the force, we define the angular momentum $$\boldsymbol{L}$$ as the rotational counterpart of momentum:
$\boldsymbol{L}=\boldsymbol{r} \times \boldsymbol{p} \label{rp}$
$\boldsymbol{L}=I \boldsymbol{\omega} \label{iomega}$
where $$I$$ is the moment of inertia of the body with respect to the symmetry axis around which it rotates. Equation \ref{iomega} also holds for a collection of particles rotating about a symmetry axis through their center of mass, as readily follows from 5.4.2 and \ref{rp}. However, it does not hold in general, as in general, $$\boldsymbol{L}$$ does not have to be parallel to $$\boldsymbol{\omega}$$. For the general case, we need to consider a moment of inertia tensor $$\boldsymbol{I}$$ (represented as a $$3×3$$ matrix) and write $$\boldsymbol{L}=\boldsymbol{I} \cdot \boldsymbol{\omega}$$. We’ll consider this case in more detail in Section 7.3.