Comparing 2 formula derivations

[Vykan12]

Here is how my book goes about defining the moment of inertia.

To begin, we think of a body as being made up of a large number of particles, with masses m_1, m_2, ... at distances r_1, r_2, from the axis of rotation. [...] The particles don't necessarily all lie in the same plane, so we specify that r_i is the perpendicular distance from the axis to the ith particle.

Now here's how my book goes about deriving an expression for the angular momentum of a rigid body.

We can use eq (10.25) to find the total angular momentum of a rigid body rating about the z axis with angular speed w. First consider a thing slice of the body lying in the xy plane.

From this the book derives that L = Iw

We can do the same calculation for other slices of the body, all parallel to the xy-plane. For points that do not lie in the xy-plane, a complication arises because the r vectors have components in the z direction as well as the x and y directions.

My question is in regards to the bolded part. How come in one derivation we assume r_i is a measure of the perpendicular distance from the axis of rotation, and in the other we made no such distinction, but rather considered r_i a 3-D vector?

 

[Vykan12]

My physics teacher answered this for me.

The only reasons the derivations are different regarding r_i is because inertia isn't a vector whereas angular momentum is.