Every solid object, such as the rods in the image below, has an inertia tensor. We can write the inertia tensor for the black rod, which keeps changing its orientation, as a sum of various multiples of the inertia tensors for the 6 fixed rods. The plot at the bottom shows how these 6 components change as the black rod points in different directions.
But why do we need 6 components? In 3D space, shouldn’t 3 be enough?
To see why we really need 6, let’s look at the inertia tensor. This is a matrix, I, that we use to multiply the angular velocity vector of an object, ω, to get the angular momentum, L.
The velocity of a point mass whose angular velocity is ω is given by:
v = ω × r
where r=(x,y,z) gives the coordinates of the point mass.
The linear momentum is:
p = m v = m ω × r
and the angular momentum is:
L = r × p = m r × (ω × r)
To get the same thing using matrices, note that we can write the cross product of r with any vector like ω as:
r × ω = C(r) ω
where C(r) is the matrix:
0 –z y
z 0 –x
–y x 0
This lets us write:
L = m r × (ω × r)
= –m r × (r × ω)
= –m C(r)^2 ω
Minus the square of the matrix C(r) is:
y^2+z^2 –xy –xz
–xy x^2+z^2 –yz
–xz –yz x^2+y^2
So the inertia tensor for a point mass is:
I = –m C(r)^2
This is the matrix above, multiplied by m.
In general, we add up (or integrate) over all the point masses in the body, to get the inertia tensor for the whole thing.
The inertia tensor we obtain this way will always be a symmetric matrix, so it can be specified with 6 numbers: 3 on the diagonal, and 3 above. So the inertia tensors belong to a 6-dimensional space.