Angular momentum without an axis of rotation?

Question

A rigid body consists of particles of masses $m_i$ at distances $r_i$ from the origin, having linear momentum $p_i$. My teacher defined $∑r_i×p_i$ as simply the angular momentum of the system. But in case of rotation of a body about an axis, he defined the exact same thing but called it the "angular momentum of the body about the axis of rotation". Why is that?

Answer 1

When you write $\sum r_i\times p_i$ you're implicitly depending on an origin point somewhere in space, from which you're measuring the $r_i$s.

So you're speaking about angular momentum about that point.

In the case of a rigid rotation it turns out that it doesn't matter which point on the axis of rotation you're using as the origin, so in that case you can just ask for angular momentum about the entire axis.